· POINCARE AND SOBOLEV INEQUALITIES FOR DIFFERENTIAL FORMS IN HEISENBERG GROUPS AND CONTACT...

POINCARE AND SOBOLEV INEQUALITIES FOR

DIFFERENTIAL FORMS IN HEISENBERG GROUPS

AND CONTACT MANIFOLDS

ANNALISA BALDI

BRUNO FRANCHI

PIERRE PANSU

Abstract. In this paper, we prove contact Poincare and Sobolev inequalities

in Heisenberg groups Hn, where the word “contact” is meant to stress thatde Rham’s exterior differential is replaced by the exterior differential of the

so-called Rumin’s complex (E•0 , dc), that recovers the scale invariance under

the group dilations associated with the stratification of the Lie algebra of Hn.In addition, we construct smoothing operators for differential forms on sub-

Riemannian contact manifolds with bounded geometry, that act trivially on

cohomology. For instance, this allows to replace a closed form, up to adding acontrolled exact form, with a much more regular differential form.

Contents

1. Introduction 21.1. Sobolev and Poincare inequalities for differential forms 21.2. Contact manifolds 31.3. Results on Poincare and Sobolev inequalities 41.4. State of the art 51.5. Open questions 61.6. Global homotopy operators 71.7. Local homotopy operators 71.8. Global smoothing 81.9. Structure of the paper 82. Heisenberg groups and Rumin’s complex (E•0 , dc) 92.1. Differential forms on Heisenberg groups 92.2. Rumin’s complex on Heisenberg groups 122.3. Rumin’s complex in contact manifolds 163. Kernels and Laplacians 183.1. Kernels in Heisenberg groups 183.2. Rumin’s Laplacians 204. Function spaces 234.1. Sobolev spaces on Heisenberg groups 234.2. Sobolev spaces on contact sub-Riemannian manifolds with bounded

geometry 265. Homotopy formulae and Poincare and Sobolev inequalities 28

1991 Mathematics Subject Classification. 58A10, 35R03, 26D15, 43A80, 53D10 46E35.Key words and phrases. Heisenberg groups, differential forms, Sobolev-Poincare inequalities,

contact manifolds, homotopy formula.1

2 ANNALISA BALDI, BRUNO FRANCHI, PIERRE PANSU

6. Contact manifolds and global smoothing 397. Large scale geometry of contact sub-Riemannian manifolds 427.1. Three-dimensional Lie groups 437.2. Other examples 437.3. Further remarks 44Acknowledgments 44References 44

1. Introduction

1.1. Sobolev and Poincare inequalities for differential forms. The Sobolevinequality in Rn states that, if u is a compactly supported function, then

‖u‖q ≤ Cp,q,n‖du‖pwhenever

1 ≤ p, q < +∞, 1

p− 1

q=

1

n,

where du is the differential of u (that is a 1-form).A local version, for functions supported in the unit ball, holds under the weaker

assumption

1 ≤ p, q < +∞, 1

p− 1

q≤ 1

n.

Poincare’s inequality is a variant for functions u defined, but not necessarilycompactly supported, in the unit ball B. It states that there exists a real numbercu such that

‖u− cu‖q ≤ Cp,q,n‖du‖p.

Alternatively, for a given exact 1-form ω on B, there exists a function u on B suchthat du = ω on B, and such that

‖u‖q ≤ Cp,q,n‖ω‖p.

This suggests the following generalization for higher degree differential forms inRiemannian manifolds.

Let M be a Riemannian manifold, with or without boundary. We say thata global Poincare inequality holds on M , if there exists a positive constant C =C(M,p, q) such that for every exact h-form ω on M , belonging to Lp, there existsa (h− 1)-form φ such that dφ = ω and

‖φ‖q ≤ C ‖ω‖p.

Shortly, we shall say that Poincarep,q(h) holds.

A global Sobolev inequality holds on M , if for every exact compactly supportedh-form ω on M , belonging to Lp, there exists a compactly supported (h− 1)-formφ such that dφ = ω and

‖φ‖q ≤ C ‖ω‖p.

Again, we shall say that Sobolevp,q(h) holds.

POINCARE AND SOBOLEV INEQUALITIES FOR DIFFERENTIAL FORMS 3

In both statements, the assumption that given forms are exact is there to separatethe topological problem (whether a given closed form is exact) from the analyticalone (whether a primitive can be upgraded to one which satisfies estimates).

For bounded convex domains, the global Poincare and Sobolev inequalities holdfor 1 < p < ∞ (see, respectively, [33], Corollary 4.2, and [42], Theorem 4.1 andequation (169)). However, for more general Euclidean domains, the validity ofPoincare inequality is sensitive to irregularities of boundaries. One way to eliminatesuch a dependence is to allow a loss on domain. For the case p = 1 in the Euclideansetting we refer to [4].

Say an interior Poincare inequality Poincarep,q(h) holds on M if for every smallenough r > 0 and large enough λ ≥ 1, there exists a constant C = C(M,p, q, r, λ)such that for every x ∈M and every exact h-form ω on B(x, λr), belonging to Lp,there exists a (h− 1)-form φ on B(x, r) such that dφ = ω on B(x, r) and

‖φ‖Lq(B(x,r)) ≤ C ‖ω‖Lp(B(x,λr)).(1)

By interior Sobolev inequalities, we mean that, if ω is supported in B(x, r), thenthere exists φ supported in B(x, λr) such that dφ = ω and

‖φ‖Lq(B(x,λr)) ≤ C ‖ω‖Lp(B(x,r)).(2)

It turns out that in several situations, the loss on domain is harmless. This isthe case for Lq,p-cohomological applications, see [47].

Let us comment on the terminology. Due to the loss on domain, inequality (1)provides no information on the behaviour of differential forms near the boundary oftheir domain of definition, this is why we speak of an interior Poincare inequality.

1.2. Contact manifolds. A contact structure on an odd-dimensional manifold Mis a smooth distribution of hyperplanes H which is maximally nonintegrable inthe following sense: if θ is a locally defined smooth 1-form such that H = ker(θ),then dθ restricts to a non-degenerate 2-form on H, i.e. if 2n + 1 is the dimensionof M , then θ ∧ (dθ)n 6= 0 on M (see [41], Proposition 3.41). A contact manifold(M,H) is the data of a smooth manifold M and a contact structure H on M .Contact diffeomorphisms (also called contactomomorphisms: see Definition 2.13)are contact structure preserving diffeomorphisms between contact manifolds. Theprototype of a contact manifold is the Heisenberg group Hn, the simply connectedLie group whose Lie algebra is the central extension

(3) h = h1 ⊕ h2, with h2 = R = Z(h),

with bracket h1 ⊗ h1 → h2 = R being a non-degenerate skew-symmetric 2-form.The contact structure is obtained by left-translating h1. According to a theorem byDarboux, every contact manifold is locally contactomorphic to Hn. The HeisenbergLie algebra admits a one parameter group of automorphisms δt,

δt = t on h1, δt = t2 on h2,

which are counterpart of the usual Euclidean dilations in Rn. Thus, differentialforms on h split into 2 eigenspaces under δt, therefore de Rham complex lacks scaleinvariance under these anisotropic dilations.

A substitute for de Rham’s complex, that recovers scale invariance under δt hasbeen defined by M. Rumin, [50]. It makes sense for arbitrary contact manifolds(M,H) and it is invariant under contactomorphisms.


Let h = 0, . . . , 2n+1. Rumin’s substitute for smooth differential forms of degreeh are the smooth sections of a vector bundle Eh0 . If h ≤ n, Eh0 is a subbundle ofΛhH∗. If h ≥ n, Eh0 is a subbundle of ΛhH∗⊗ (TM/H). Rumin’s substitute for deRham’s exterior differential is a linear differential operator dc from sections of Eh0to sections of Eh+1

0 such that d2c = 0.

We stress that the operator dc has order 2 when h = n and order 1otherwise.

This phenomenon will be a major issue in the proofs of our results and will affectthe choice of the exponents p, q in our inequalities

The data of (M,H) equipped with a scalar product g, defined on sub-bundleH only, is called a sub-Riemannian contact manifold and we shall write (M,H, g).The scalar product on H determines a choice of a local contact form θ, hence anorm on the line bundle TM/H. Therefore Eh0 are endowed with a scalar product.Using θ ∧ (dθ)n as a volume form, one gets Lp-norms on spaces of smooth Rumindifferential forms.

In any sub-Riemannian contact manifold (M,H, g) we can define a sub-Rieman-nian distance dM (see e.g. [43]) inducing on M the same topology of M as amanifold. In particular, Heisenberg groups Hn can be viewed as sub-Riemanniancontact manifolds. If we choose on the contact sub-bundle of Hn a left-invariantmetric, it turns out that the associated sub-Riemanian metric is also left-invariant.It is customary to call this distance in Hn a Carnot-Caratheodory distance.

Poincare and Sobolev inequalities for differential forms make sense on contactsub-Riemannian manifolds: merely replace the exterior differential d with dc. Allleft-invariant sub-Riemannian metrics on Heisenberg group are bi-Lipschitz equiva-lent, hence we may refer to sub-Riemannian Heisenberg group without referring toa specific left-invariant metric: if a Poincare inequality holds for some left-invariantmetric, it holds for all of them. On the other hand, in absence of symmetry as-sumptions, large scale behaviors of sub-Riemannian contact manifolds are diverse.Examples illustrating this phenomenon will be given in Section 7.

1.3. Results on Poincare and Sobolev inequalities. In this paper, we proveglobal H-Poincare and H-Sobolev inequalities and interior H-Poincare and H-Sobolev inequalities in Heisenberg groups, where the prefix H is meant to stressthat the exterior differential is replaced with Rumin’s exterior differential dc. Therange of parameters differs slightly from the Euclidean case, due to the fact that dchas order 2 in middle dimension. Let h ∈ 0, . . . , 2n+ 1. We say that assumptionE(h, p, q, n) holds if 1 < p ≤ q <∞ satisfy

1

p− 1

q=

1

2n+2 if h 6= n+ 1,2

2n+2 if h = n+ 1.

Say that assumption I(h, p, q, n) holds if 1 < p ≤ q <∞ satisfy

1

p− 1

q≤

1

2n+2 if h 6= n+ 1,2

2n+2 if h = n+ 1.

Theorem 1.1. Under assumption E(h, p, q, n), global H-Poincarep,q and H-Sobolevp,qinequalities hold for Rumin’s h-forms on Hn.


Theorem 1.2. Under assumption I(h, p, q, n), interior H-Poincarep,q and interiorH-Sobolevp,q inequalities hold for Rumin’s h-forms on Hn.

Precise formulations of interior Poincare and Sobolev inequalities are given insection 5.

Remark 1.3. We stress that the core of the present paper is the proof of theinterior inequalities of Theorem 1.2. In fact, since p > 1, the global estimates ofTheorem 1.1 are more or less straightforward consequences of the Lp−Lq continuityof singular integrals of potential type (see Section 1.6 below).

Here is a simple consequence of these results. Combining both theorems withresults from [47], we get

Corollary 1.4. Under assumption E(h, p, q, n), the `q,p-cohomology in degree h ofHn vanishes.

Our third result is the construction of a smoothing homotopy on general contactmanifolds. Under a bounded geometry assumption, uniform estimates can be given(precise definitions of bounded geometry contact manifolds, as well as of associatedSobolev spaces W j,p, will be given in Section 4.2).

Theorem 1.5. Let k ≥ 3 be an integer. Let (M,H, g) be a 2n + 1-dimensionalsub-Riemannian contact manifold of bounded Ck-geometry. Under assumptionI(h, p, q, n), there exist operators SM and TM on h-forms on M which are boundedfrom W j,p to W j,q for all 0 ≤ j ≤ k − 1, and such that

(4) 1 = SM + dcTM + TMdc.

Furthermore, SM and TM are bounded from W j−1,p to W j,p if j ≥ 1 (resp. fromW j−2,p to W j,p if j ≥ 2 and degree h = n+ 1).

We stress that the “approximate homotopy formula” (4) has no consequences forthe cohomology of M . The iteration of the process yields an operator SM which isbounded from Lp to W k−1,q, and still acts trivially on cohomology. For instance,it is possible to replace a closed form with a much more regular differential form(up to adding a controlled exact form).

1.4. State of the art. This paper is part of a larger project aimed to prove (p, q)-Poincare and Sobolev inequalities in Heisenberg groups when 1 ≤ p < q ≤ ∞. Thusit seems convenient to point out the different cases we have to deal with. Let usrestrict ourselves for a while to Euclidean spaces Rn and Heisenberg groups Hn.The first fundamental distinction is the following:

i) global inequalities (i.e. inequalities on all the space Rn or Hn);ii) interior inequalities (for instance on Carnot-Caratheodory balls).

For each one of the above geometric assumptions we must distinguish between

iii) the case p = 1;iv) the case p > 1.

In the scalar case, (p, q)-Poincare and Sobolev inequalities are well understoodboth in Euclidean spaces and in Heisenberg groups for all p ≥ 1. Consider nowdifferential forms of higher degree.

For the case p = 1, global inequalities in Rn (Gagliardo-Nirenberg inequalitiesfor differential forms) have been proved by Bourgain & Brezis ([15]) and Lanzani


& Stein ([36]) via a suitable identity for closed differential forms and relying oncareful estimates for divergence-free vector fields. Thanks to the counterpart ofthis identity proved by Chanillo & van Schaftingen in homogeneous groups ([18]),similar global inequalites for differential forms in Hn were proved in [3]. We stressthat in [3] algebra plays an important role precisely in the proof of the identitiesfor closed forms, therefore apart from Heisenberg groups, only a handful of moregeneral nilpotent groups have been treated, [11].

Interior inequalities when p = 1 use the estimate of [3] combined with an ap-proximate homotopy formula introduced in the present paper, but require a newdifferent argument to control the commutator between Rumin’s exterior differential(or de Rham’s exterior differential in Rn) and multiplication by a cut-off function.These inequalities are proved for Heisenberg groups in [6] and in [4] for Euclideanspaces. Notice that in the Heisenberg group case, one more algebraic obstacle showsup, averages of L1 forms, see [49].

Consider now the case p > 1. In the Euclidean setting, interior Poincare in-equalities for p > 1 are proved in [33]. However, the arguments of [33] do notextend to Heisenberg groups. Thus, the core of the present paper is the proof ofinterior Poincare and Sobolev inequalities in Hn when p > 1. Indeed, as we shallpoint out later (see Remark 1.3), when p > 1 global inequalities in Hn (as well asin Rn) are more or less straightforward.

On the contrary, interior inequalities require a different more sophisticated ar-gument (see Section 1.7 for a gist of our proof). At the same time, the techniquesintroduced in the present paper differ substantially from those of [3] for globalinequalities for p = 1.

The case when q = ∞ can be obtained by duality, and this will appear in [5].We refer also to [7] for endpoint inequalities in Orlicz spaces.

For more general sub-Riemannian spaces, the strategy is to reduce to large scaleinvariants (see section 7). For this, one must pass via interior inequalities and aglobal smoothing procedure, like in Theorems 1.2 and 1.5. In particular, in thepresent paper and in [6] we deal with a special class of sub-Riemannian manifolds,the sub-Riemannian contact manifold of bounded Ck-geometry as in Definition 4.9.

1.5. Open questions. Keeping in mind the analogous inequalities in the scalarcase, the following (still open) questions naturally arise.

1. Do Poincare and Sobolev inequalities hold without loss of the domain forsome family of specific domains as, e.g., for metric balls associated with aleft-invariant homogeneous distance?

2. Since Heisenberg groups provide the simplest non-commutative instance ofarbitrary Carnot groups (connected, simply connected stratified nilpotentgroups: see [45]), the following question naturally arises: how much of theseresults do extend to more general Carnot groups?

Let us make a few comments about the previous questions.

1. When dealing with scalar functions it is possible to obtain H-Poincarep,qinequalities on Carnot-Caratheodory balls without loss on the domain andthe argument relies on the so-called Boman chain condition (see, e.g. [22],[24]). However, it is not clear at all how to extend this technique to differ-ential forms.


2. The argument used in this paper relies on an appropriate approximatehomotopy formula (see point (2) in Section 1.7 below). It is reasonable toexpect that the construction of the approximate homotopy operator couldbe generalized to more general Carnot groups using the construction carriedout in [8] and [48] to prove a compensated compactness result (see formulæ(37) and (38) in [8]). However, for Carnot groups we expect only unsharpestimates, due to the crucial role of a fundamental solution of a 0-orderLaplace operator mixing up components of forms of different homogeneity.Further comments related to this question can be found in Remark 5.22below where specific examples in more general Carnot groups are given.

Let us give now a sketch of the proofs.

1.6. Global homotopy operators. The most efficient way to prove a Poincareinequality is to find a homotopy between identity and 0 on the complex of differentialforms, i.e. a linear operator K that raises the degree by 1 and satisfies

I = dK +Kd.

More generally, we shall deal with homotopies between identity and other operatorsP , i.e. of the form

I − P = dK +Kd.

In Euclidean space, the Laplacian provides us such a homotopy. Write ∆ =dδ+ δd. Denote by ∆−1 the operator of convolution with the fundamental solutionof the Laplacian. Then ∆−1 commutes with d and its adjoint δ, hence KEuc = δ∆−1

satisfies I = dKEuc +KEucd on globally defined Lp differential forms. Furthermore,KEuc is bounded Lp → Lq provided 1

p−1q = 1

n . This proves the global Poincarep,q(h)

inequality for Euclidean space.Rumin defines a Laplacian ∆c by ∆c = dcδc + δcdc when both dc and δc are

first order horizontal differential operators, and by ∆c = (dcδc)2 + δcdc or ∆c =

dcδc + (δcdc)2 near middle dimension (i.e. when h = n or h = n+ 1, respectively),

when one of them has order 2. This leads to a homotopy of the form K0 = δc∆−1c or

K0 = δcdcδc∆−1c depending on degree. Again, K0 is a singular integral of potential

type associated with a homogeneous kernel and therefore is bounded from Lp to Lq

under assumption E(h, p, q, n) (see [20] or [21] for the continuity of Riesz potentialsin homogeneous groups). This proves the global H-Poincarep,q(h) inequality forHeisenberg group, Theorem 1.1.

1.7. Local homotopy operators. We pass to interior estimates. In Euclideanspace, Poincare’s Lemma asserts that every closed form on a ball is exact. We needa quantitative version of this statement. The standard proof of Poincare’s Lemmarelies on a homotopy operator which depends on the choice of an origin. Averagingover origins yields a bounded operator KEuc : Lp → Lq, as was observed by Iwaniecand Lutoborski, [33]. This proves the global Euclidean Poincarep,q(h) inequality forconvex Euclidean domains. A support preserving variant JEuc : Lp → Lq appearsin Mitrea-Mitrea-Monniaux, [42] and this proves the global Euclidean Sobolevp,qinequality for bounded convex Euclidean domains. Incidentally, since for ballsconstants do not depend on the radius of the ball, this reproves the global EuclideanSobolevp,q inequality for Euclidean spaces.


In this paper a sub-Riemannian counterpart is obtained using the homotopyequivalence of de Rham’s and Rumin’s complexes. Since this homotopy is a differ-ential operator, a preliminary smoothing operation is needed. This is obtained bylocalizing (multiplying the kernel with cut-offs) the global homotopy K0 providedby the inverse of Rumin’s (modified) Laplacian.

Hence the proof goes as follows (see Section 5):

(1) Show that the inverse K0 of Rumin’s modified Laplacian on all of Hn isgiven by a homogeneous kernel k0. Deduce bounds Lp → W 1,q, where q, pare as above. Conclude that K0 is an exact homotopy for globally definedLp forms. Basically, this step does not contain any new idea, relying onlyon the estimates of the fundamental solution of Rumin’s modified Laplacian(see [10]) and on classical estimates for convolution kernels in homogeneousgroups (see [20], [21]).

(2) Take a smooth cut-off function ψ, ψ ≡ 1 in a neighborhood of the origin,and split k0 = ψk0 + (1 − ψ)k0, so that ψk0 has small support near theorigin and (1 − ψ)k0 is smooth. Denote by T the convolution operatorassociated with the kernel ψk0, and by Ksmooth the convolution operatorassociated with the kernel (1 − ψ)k0. It turns out that T is a homotopyon balls (with a loss on domain) between the identity I and the operatorS := dcKsmooth +Ksmoothdc (which is smoothing), i.e. I − S = dcT + Tdc.The operator S provides the required local smoothing operator.

(3) Compose Iwaniec & Lutoborski’s averaged Poincare homotopy for the deRham complex and Rumin’s homotopy, and apply the result to smoothedforms. This proves an interior Poincare inequality in Heisenberg groups.Replacing Iwaniec & Lutoborski’s homotopy with Mitrea, Mitrea & Mon-niaux’s homotopy leads to an interior Sobolev inequality in Heisenberggroups.

1.8. Global smoothing. Now we piece together local homotopy operators intoglobally defined smoothing operators. Let k ≥ 3. Let (M,H, g) be a boundedCk-geometry sub-Riemannian contact manifold. Pick a uniform covering by equalradius balls. Let χj be a partition of unity subordinate to this covering. Let φjbe the corresponding charts from the unit Heisenberg ball. Let Sj and Tj denotethe smoothing and homotopy operators associated with φj using the pull-backoperator. Set

T =∑j

Tjχj , S =∑j

Sjχj + Tj [χj , dc].

When dc is first order, the commutator [χj , dc] is an order 0 differential operator,hence Tj [χj , dc] gains 1 derivative. When dc is second order, [χj , dc] is a first orderdifferential operator. It turns out that precisely in this case, Tj gains 2 derivatives,hence Tj [χj , dc] gains 1 derivative in this case as well.

The details are discussed in Section 6.

1.9. Structure of the paper. In Section 2 we collect basic results about Heisen-berg groups Hn and differential forms in Hn. Successively, we remind the notionof Rumin’s complex for Heisenberg groups as well as for general contact mani-folds, providing explicit examples in low dimensions. In Section 3 we present alist of general results for Folland-Stein homogeneous kernels, and, in particular,


for matrix-valued kernels associated with Rumin’s homogeneous Laplacian in Hn.Section 4 is devoted to theory of Folland-Stein Sobolev spaces in Heisenberg groupsand in sub-Riemannian contact manifolds with bounded geometry. In particular,in Section 4.2 we precise the notion and the properties of manifolds with boundedgeometry. Section 5 is the core of the paper, containing an approximate homotopyformulae (i.e. and homotopy formula with a smoothing error term) and Poincareand Sobolev inequalities for differential forms in Hn. Then, in Section 6 we areable to prove a similar approximate homotopy formula for sub-Riemannian con-tact manifolds with bounded geometry. The error term is a regularizing operatorwith “maximal regularity”. Finally, Section 7 contains a few examples of contactmanifolds with bounded geometry, and a brief discussion of the `q,p cohomology.

2. Heisenberg groups and Rumin’s complex (E•0 , dc)

2.1. Differential forms on Heisenberg groups. We denote by Hn the (2n+1)-dimensional Heisenberg group, identified with R2n+1 through exponential coordi-nates. A point p ∈ Hn is denoted by p = (x, y, t), with both x, y ∈ Rn and t ∈ R.If p and p′ ∈ Hn, the group operation is defined by

p · p′ = (x+ x′, y + y′, t+ t′ + 2

n∑j=1

(xjy′j − yjx′j)).

Notice that Hn can be equivalently identified with C× R endowed with the groupoperation

(z, t) · (ζ, τ) := (z + ζ, t+ τ + 2 Im (zζ)).

The unit element of Hn is the origin, that will be denote by e. For any q ∈ Hn,the (left) translation τq : Hn → Hn is defined as

p 7→ τqp := q · p.

For a general review on Heisenberg groups and their properties, we refer to [56],[30] and to [57]. We limit ourselves to fix some notations, following [26].

First we notice that Heisenberg groups are smooth manifolds (and therefore areLie groups). In particular, the pull-back of differential forms is well defined asfollows (see, e.g. [28], Proposition 1.106);

Definition 2.1. If U ,V are open subsets of Hn, and f : U → V is a diffeomor-phism, then for any differential form α of degree h, we denote by f ]α the pull-backform in U defined by

(f ]α)(p)(v1, . . . , vh) := α(f(p))(df(p)v1, . . . , df(p)vh)

for any h-tuple (v1, . . . , vh) of tangent vectors at p.

The Heisenberg group Hn can be endowed with the homogeneous norm (Cygan-Koranyi norm): if p = (x, y, t) ∈ Hn, then we set

(5) %(p) =((x2 + y2)2 + 16t2

)1/4,

and we define the gauge distance (a true distance, see [56], p. 638), that is leftinvariant i.e. d(τqp, τqp

′) = d(p, p′) for all p, p′ ∈ Hn) as

(6) d(p, q) := %(p−1 · q).


Notice that d is equivalent to the Carnot-Caratheodory distance on Hn (see, e.g.,[14], Corollary 5.1.5). Finally, the balls for the metric d are the so-called Cygan-Koranyi balls

(7) B(p, r) := q ∈ Hn; d(p, q) < r.

Notice that Cygan-Koranyi balls are convex smooth sets.A straightforward computation shows that, if ρ(p) < 1, then

(8) |p| ≤ ρ(p) ≤ |p|1/2.

It is well known that the topological dimension of Hn is 2n+1, since as a smoothmanifold it coincides with R2n+1, whereas the Hausdorff dimension of (Hn, d) isQ := 2n+ 2 (the so called homogeneous dimension of Hn).

We denote by h the Lie algebra of the left invariant vector fields of Hn. Thestandard basis of h is given, for i = 1, . . . , n, by

Xi := ∂xi− 2yi∂t, Yi := ∂yi + 2xi∂t, T := ∂t.

The only non-trivial commutation relations are [Xj , Yj ] = 4T , for j = 1, . . . , n. Thehorizontal subspace h1 is the subspace of h spanned by X1, . . . , Xn and Y1, . . . , Yn:h1 := span X1, . . . , Xn, Y1, . . . , Yn .Coherently, from now on, we refer to X1, . . . , Xn, Y1, . . . , Yn (identified with firstorder differential operators) as the horizontal derivatives. Denoting by h2 the linearspan of T , the 2-step stratification of h is expressed by

h = h1 ⊕ h2.

The stratification of the Lie algebra h induces a family of non-isotropic dilationsδλ : Hn → Hn, λ > 0 as follows: if p = (x, y, t) ∈ Hn, then

(9) δλ(x, y, t) = (λx, λy, λ2t).

The vector space h can be endowed with an inner product, indicated by 〈·, ·〉,making X1, . . . , Xn, Y1, . . . , Yn and T orthonormal.

Throughout this paper, we write also

(10) Wi := Xi, Wi+n := Yi and W2n+1 := T, for i = 1, . . . , n.

The dual space of h is denoted by∧1

h. The basis of∧1

h, dual to the basisX1, . . . , Yn, T, is the family of covectors dx1, . . . , dxn, dy1, . . . , dyn, θ where

(11) θ := dt− 2

n∑j=1

(xjdyj − yjdxj)

is called the contact form in Hn. We denote by 〈·, ·〉 the inner product in∧1

h thatmakes (dx1, . . . , dyn, θ) an orthonormal basis.

Coherently with the previous notation (10), we set

ωi := dxi, ωi+n := dyi and ω2n+1 := θ, for i = 1, . . . , n.

We put∧

0 h :=∧0

h = R and, for 1 ≤ h ≤ 2n+ 1,∧hh := spanωi1 ∧ · · · ∧ ωih : 1 ≤ i1 < · · · < ih ≤ 2n+ 1.


In the sequel we shall denote by Θh the basis of∧h

h defined by

Θh := ωi1 ∧ · · · ∧ ωih : 1 ≤ i1 < · · · < ih ≤ 2n+ 1.To avoid cumbersome notations, if I := (i1, . . . , ih), we write

ωI := ωi1 ∧ · · · ∧ ωih .

The inner product 〈·, ·〉 on∧1

h yields naturally an inner product 〈·, ·〉 on∧h

hmaking Θh an orthonormal basis.

The volume (2n+ 1)-form ω1 ∧ · · · ∧ ω2n+1 will be also written as dV .

Throughout this paper, the elements of∧h

h are identified with left invariantdifferential forms of degree h on Hn.

Definition 2.2. A h-form α on Hn is said left invariant if

τ#q α = α for any q ∈ Hn.

The same construction can be performed starting from the vector subspace h1 ⊂h, obtaining the horizontal h-covectors

∧hh1 := spanωi1 ∧ · · · ∧ ωih : 1 ≤ i1 < · · · < ih ≤ 2n.

It is easy to see that

Θh0 := Θh ∩

∧hh1

provides an orthonormal basis of∧h

h1.Keeping in mind that the Lie algebra h can be identified with the tangent space

to Hn at x = e (see, e.g. [28], Proposition 1.72), starting from∧h

h we can define

by left translation a fiber bundle over Hn that we can still denote by∧h

h. We

can think of h-forms as sections of∧h

h. We denote by Ωh the vector space of allsmooth h-forms.

We already pointed out in Section 1.2 that the stratification of the Lie algebra hyields a lack of homogeneity of de Rham’s exterior differential with respect to groupdilations δλ. Thus, to keep into account the different degrees of homogeneity of thecovectors when they vanish on different layers of the stratification, we introducethe notion of weight of a covector as follows.

Definition 2.3. If η 6= 0, η ∈∧1

h1, we say that η has weight 1, and we write

w(η) = 1. If η = θ, we say w(η) = 2. More generally, if η ∈∧h

h, η 6= 0, we saythat η has pure weight p if η is a linear combination of covectors ωi1 ∧ · · · ∧ ωihwith w(ωi1) + · · ·+ w(ωih) = p.

Notice that, if η, ζ ∈∧h

h and w(η) 6= w(ζ), then 〈η, ζ〉 = 0 (see [8], Remark2.4). We notice also that w(dθ) = w(θ).

We stress that generic covectors may fail to have a pure weight: it is enough toconsider H1 and the covector dx1 + θ ∈

∧1h. However, the following result holds

(see [8], formula (16)):

(12)∧h

h =∧h,h

h⊕∧h,h+1

h =∧h

h1 ⊕(∧h−1

h1

)∧ θ,

where∧h,p

h denotes the linear span of the h-covectors of weight p. By our previousremark, the decomposition (12) is orthogonal. In addition, since the elements of


the basis Θh have pure weights, a basis of∧h,p

h is given by Θh,p := Θh ∩∧h,p

h(such a basis is usually called an adapted basis).

We notice that, according to (12), the weight of a h-form is either h or h + 1and there are no forms of weight h + 2, since there is only one 1-form of weight2. Something analogous can be possible for instance in Hn × R, but it fails to bepossible already in the case of general step 2 groups with higher dimensional center(see also Remark 5.22 below).

As above, starting from∧h,p

h, we can define by left translation a fiber bundle

over Hn that we can still denote by∧h,p

h. Thus, if we denote by Ωh,p the vectorspace of all smooth h–forms in Hn of weight p, i.e. the space of all smooth sections

of∧h,p

h, we have

(13) Ωh = Ωh,h ⊕ Ωh,h+1.

2.2. Rumin’s complex on Heisenberg groups. Let us give a short introductionto Rumin’s complex. For a more detailed presentation we refer to Rumin’s papers[53]. Here we follow the presentation of [8]. The exterior differential d does notpreserve weights. It splits into

d = d0 + d1 + d2

where d0 preserves weight, d1 increases weight by 1 unit and d2 increases weight by2 units.

More explicitly, let α ∈ Ωh be a (say) smooth form of pure weight h. We canwrite

α =∑

ωI∈Θh0

αI ωI , with αI ∈ C∞(Hn).

Then

dα =∑

ωI∈Θh0

2n∑j=1

(WjαI)ωj ∧ ωI +∑

ωI∈Θh0

(TαI) θ ∧ ωI = d1α+ d2α,

and d0α = 0. On the other hand, if α ∈ Ωh,h+1 has pure weight h+ 1, then

α =∑

ωJ∈Θh−10

αJ θ ∧ ωJ ,

and

dα =∑

ωJ∈Θh0

αJ dθ ∧ ωJ +∑

ωJ∈Θh0

2n∑j=1

(WjαJ)ωj ∧ θ ∧ ωI = d0α+ d1α,

and d2α = 0.It is crucial to notice that d0 is an algebraic operator, in the sense that for any

real-valued f ∈ C∞(Hn) we have

d0(fα) = fd0α,

so that its action can be identified at any point with the action of a linear operator

from∧h

h to∧h+1

h (that we denote again by d0).Following M. Rumin ([53], [51]) we give the following definition:


Definition 2.4. If 0 ≤ h ≤ 2n + 1, keeping in mind that∧h

h is endowed with acanonical inner product, we set

Eh0 := ker d0 ∩ (Im d0)⊥.

Straightforwardly, Eh0 inherits from∧h

h the inner product.

As above, E•0 defines by left translation a fibre bundle over Hn, that we stilldenote by E•0 . To avoid cumbersome notations, we denote also by E•0 the space ofsections of this fibre bundle.

Let L :∧h

h→∧h+2

h the Lefschetz operator defined by

(14) Lξ = dθ ∧ ξ.

Then the spaces E•0 can be defined explicitly as follows:

Theorem 2.5 (see [50], [52]). We have:

i) E10 =

∧1h1;

ii) if 2 ≤ h ≤ n, then Eh0 =∧h

h1 ∩(∧h−2

h1 ∧ dθ)⊥

(i.e. Eh0 is the space of

the so-called primitive covectors of∧h

h1);

iii) if n < h ≤ 2n + 1, then Eh0 = α = β ∧ θ, β ∈∧h−1

h1, γ ∧ dθ = 0 =θ ∧ kerL;

iv) if 1 < h ≤ n, then Nh := dimEh0 =(

2nh

)−(

2nh−2

);

v) if ∗ denotes the Hodge duality associated with the inner product in∧•

h and

the volume form dV , then ∗Eh0 = E2n+1−h0 .

Notice that all forms in Eh0 have weight h if 1 ≤ h ≤ n and weight h + 1 ifn < h ≤ 2n+ 1.

A further geometric interpretation (in terms of decomposition of h and of graphswithin Hn) can be found in [27].

Notice that there exists a left invariant orthonormal basis

(15) Ξh0 = ξh1 , . . . , ξhdimEh0

of Eh0 that is adapted to the filtration (12). Such a basis is explicitly constructedby induction in [1].

The core of Rumin’s theory consists in the construction of a suitable “exteriordifferential” dc : Eh0 → Eh+1

0 making E0 := (E•0 , dc) a complex homotopic to the deRham complex.

Let us sketch Rumin’s construction: first the next result (see [8], Lemma 2.11for a proof) allows us to define a (pseudo) inverse of d0 :

Lemma 2.6. If 1 ≤ h ≤ n, then ker d0 =∧h

h1. Moreover, if β ∈∧h+1

h, then

there exists a unique γ ∈∧h

h ∩ (ker d0)⊥ such that

d0γ − β ∈ R(d0)⊥.

With the notations of the previous lemma, we set

γ := d−10 β.

We notice that d−10 preserves the weights.

The following theorem summarizes the construction of the intrinsic differentialdc (for details, see [53] and [8], Section 2) .


Theorem 2.7. The de Rham complex (Ω•, d) splits into the direct sum of twosub-complexes (E•, d) and (F •, d), with

E := ker d−10 ∩ ker(d−1

0 d) and F := R(d−10 ) +R(dd−1

0 ).

Let ΠE be the projection on E along F (that is not an orthogonal projection). Wehave

i) If γ ∈ Eh0 , then• ΠEγ = γ − d−1

0 d1γ if 1 ≤ h ≤ n;• ΠEγ = γ if h > n.

ii) ΠE is a chain map, i.e.

dΠE = ΠEd.

iii) Let ΠE0be the orthogonal projection from

∧∗h on E•0 , then

(16) ΠE0= I − d−1

0 d0 − d0d−10 , ΠE⊥0

= d−10 d0 + d0d

−10 .

iv) ΠE0ΠEΠE0

= ΠE0and ΠEΠE0

ΠE = ΠE.

Set now

dc = ΠE0dΠE : Eh0 → Eh+1

0 , h = 0, . . . , 2n.

We have:

v) d2c = 0;

vi) the complex E0 := (E•0 , dc) is homotopic to the de Rham complex;

vii) dc : Eh0 → Eh+10 is a homogeneous differential operator in the horizontal

derivatives of order 1 if h 6= n, whereas dc : En0 → En+10 is an homogeneous

differential operator in the horizontal derivatives of order 2.

To illustrate the previous construction, let us write explicitly the classes Eh0 and

the differential dc : Eh0 → Eh+10 in H1 and H2 (for proofs, see e.g. [2]).

Example 2.8. Consider the first Heisenberg group H1 ≡ R3 with variables (x, y, t).With the notations of (11), we have:

E10 = span dx, dy;

E20 = span dx ∧ θ, dy ∧ θ;

E30 = span dx ∧ dy ∧ θ.

Thus, if α = α1dx+ α2dy ∈ E10 , then

a) dcα = (X2α2 − 2XY α1 + Y Xα1)dx∧ θ+ (2Y Xα2 − Y 2α1 −XY α2)dy ∧ θb) d∗cα = −(Xα1 + Y α2).

On the other hand, if α = α13dx ∧ θ + α23dy ∧ θ ∈ E20 , then

c) dcα = (Xα23 − Y α13) dx ∧ dy ∧ θ;d) d∗cα = (XY α13 − 2Y Xα13 − Y 2α23)dx+ (X2α13 + 2XY α23 − Y Xα23)dy.

Example 2.9. Choose now H2 ≡ R5, with variables (x1, x2, y1, y2, t).In this case

E10 = span dx1, dx2, dy1, dy2;

E20 = span dx1 ∧ dx2, dx1 ∧ dy2, dx2 ∧ dy1, dy1 ∧ dy2,

1√2

(dx1 ∧ dy1 − dx2 ∧ dy2).


The classes E30 and E4

0 are easily written by Hodge duality:

E30 = span dy1 ∧ dy2 ∧ θ, dx2 ∧ dy1 ∧ θ, dx1 ∧ dy2 ∧ θ, dx1 ∧ dx2 ∧ θ,

1√2

(dx1 ∧ dy1 − dx2 ∧ dy2) ∧ θ;

E40 = span dx2 ∧ dy1 ∧ dy2 ∧ θ, dx1 ∧ dy1 ∧ dy2 ∧ θ, dx1 ∧ dx2 ∧ dy2 ∧ θ,

dx1 ∧ dx2 ∧ dy1 ∧ θE5

0 = span dx1 ∧ dx2 ∧ dy1 ∧ dy2 ∧ θ = dV .

Thus, if α = α1dx1 + α2dx2 + α3dy1 + α4dy2 ∈ E10 , we have

(a) dcα = (X1α2 −X2α1)dx1 ∧ dx2 + (Y1α4 − Y2α3)dy1 ∧ dy2

+ (X1α4 − Y2α1)dx1 ∧ dy2 + (X2α3 − Y1α2)dx2 ∧ dy1

+X1α3 − Y1α1 −X2α4 + Y2α2√

2

1√2

(dx1 ∧ dy1 − dx2 ∧ dy2).

(b) δcα = −(X1α1 +X2α2 + Y1α3 + Y2α4).

Finally, if

α =α1dx1 ∧ dx2 + α2dx1 ∧ dy2 + α3dx2 ∧ dy1 + α4dy1 ∧ dy2

+α5√

2(dx1 ∧ dy1 − dx2 ∧ dy2) ∈ E2

0 ,

we have

(c) dcα =(Y 2

1 α2 − Y 22 α3 + (X1Y1 − 2Y1X1 − Y2X2)α4 −

√2Y1Y2α5

)dy1 ∧ dy2 ∧ θ

+(− Y 2

1 α1 + (X1Y1 − 2Y1X1 +X2Y2)α3 +X22α4 +

√2X2Y1α5

)dx2 ∧ dy1 ∧ θ

+(Y 2

2 α1 + (2X1Y1 − Y1X1 − Y2X2)α2 −X21α4 −

√2X1Y2α5

)dx1 ∧ dy2 ∧ θ

+((2X1Y1 − Y1X1 − 2X2Y2)α1 +X2

2α2 +X21α3 −

√2X1X2α5

)dx1 ∧ dx2 ∧ θ

+(2√

2Y1Y2α1 − 2√

2X2Y1α2 + 2√

2X1Y2α3 + 2√

2X1X2α4 + 3Tα5

)· 1√

2(dx1 ∧ dy1 − dx2 ∧ dy2) ∧ θ.

(d) δcα = (X2α1 + Y2α2 +1√2Y1α5)dx1 + (−X1α1 + Y1α3 −

1√2Y2α5)dx2

+ (−X2α3 + Y2α4 −1√2X1α5)dy1 + (−X1α2 − Y1α4 +

1√2X2α5)dy2.

Remark 2.10. The construction of Rumin’s complex can be carried out in gen-eral Carnot groups following verbatim the construction presented in Section 2.2 forHeisenberg groups, once a general notion of weight is provided. This can be easilydone in term of homogeneity of a covector with respect to group dilations (see [54],[53], [8]).

Since the exterior differential dc on Eh0 can be written in coordinates as a left-invariant homogeneous differential operator in the horizontal variables, of order 1if h 6= n and of order 2 if h = n, the proof of the following Leibniz’ formula is easy.

Lemma 2.11. If ζ is a smooth real function, then


• if h 6= n, then on Eh0 we have:

[dc, ζ] = Ph0 ,

where Ph0 : Eh0 → Eh+10 is a linear homogeneous differential operator of

degree zero, with coefficients depending only on the horizontal derivativesof ζ;• if h = n, then on En0 we have

[dc, ζ] = Pn1 + Pn0 ,

where Pn1 : En0 → En+10 is a linear homogeneous differential operator of

degree 1, with coefficients depending only on the horizontal derivatives of ζ,and where Ph0 : En0 → En+1

0 is a linear homogeneous differential operatorin the horizontal derivatives of degree 0 with coefficients depending only onsecond order horizontal derivatives of ζ.

The next remarkable property of Rumin’s complex is its invariance under contacttransformations. Here we state a special case before developing this point in section2.3 (see [9], Proposition 3.19 for a proof).

Proposition 2.12. If we write a form α =∑j αjξ

hj in coordinates with respect to

a left-invariant basis of Eh0 (see (15)) we have:

(17) τ#q α =

∑j

(αj τq)ξhj

for all q ∈ Hn. In addition, for t > 0,

(18) δ#t α = th

∑j

(αj δt)ξhj if 1 ≤ h ≤ n

and

(19) δ#t α = th+1

∑j

(αj δt)ξhj if n+ 1 ≤ h ≤ 2n+ 1 .

2.3. Rumin’s complex in contact manifolds. Let us start with the followingdefinition (see [41], Section I-3).

Definition 2.13. If (M1, H1) and (M2, H2) are contact manifolds with Hi = kerαi(i.e. αi are contact forms) i = 1, 2, U1 ⊂ M1, U2 ⊂ M2 are open sets and f is adiffeomorphism from U1 onto U2, then f is said a contact diffeomorphisms if thereexists a non-vanishing real function τ defined in U1 such that

f#α2 = τα1 in U1.

We recall that, by a classical theorem of Darboux, any contact manifold (M,H)is locally contact diffeomorphic to the Heisenberg group Hn (see [41], p. 112).

Rumin’s intrinsic complex is invariantly defined for general contact manifolds(M,H). Although the operators d0 and d−1

0 are not invariantly defined, the sub-spaces E and F of differential forms, the operator ΠE onto E parallel to F , thevectorbundles Eh0 and the projector ΠE0

are contact invariants. To see this, let usfollow [54].

Locally, H is the kernel of a smooth contact 1-form θ. Let L :∧•

H∗ →∧•

H∗

denote multiplication by dθ|H (recall (14)).Let us start with E and F . It is well known that, for every h ≤ n − 1,

Ln−h :∧h

H∗ →∧2n−h

H∗ is an isomorphism. It follows that ker(Ln−h+1) is


a complement of R(L) in∧h

H∗, if h ≤ n, and that R(L) =∧h

H∗ if h ≥ n + 1.Therefore we set

V h =

α ∈ T ∗M ; Ln−h+1(α|H) = 0 if h ≤ n,α ∈ T ∗M ; α|H = 0 otherwise.

Similarly, R(Lh−n+1) is a complement of ker(L) in∧h

H∗ if h ≥ n, and ker(L) =

0 in∧h

H∗ if h ≤ n− 1. Therefore we set

Wh =

α ∈ T ∗M ; α|H = 0 if h ≤ n− 1,

α ∈ T ∗M ; α ∈ θ ∧R(Lh−n+1) otherwise.

Changing θ to an other smooth 1-form θ′ = fθ with kernel H does not change V andW . With these choices, spaces of smooth sections of V and W (which we still denoteby V and W ) depend only on the plane field H. We can define subspaces of smoothdifferential forms E = V ∩d−1V and F = W +dW and the projector ΠE . Since noextra choices are involved, E, F and ΠE are invariant under contactomorphisms.On Heisenberg group, one recovers the spaces E and F defined in Theorem 2.7.

Next we define the sub-bundles Eh0 . In degrees h ≥ n + 1, Eh0 = θ ∧ (∧h

H∗ ∩ker(L)) is a contact invariant. Since

(ΠE0)|E = ((ΠE)|E0

)−1,

the operator dc = ((ΠE)|E0)−1 d (ΠE)|E0

is a contact invariant.In degrees h ≤ n, the restriction of differential forms to H is an isomorphism of

Eh0 to E′0h

:=∧h

H∗ ∩ ker(Ln−h+1). We note that for a differential form ω suchthat ω|H ∈ E′0, ΠE(ω) only depends on ω|H . It follows that (ΠE)|E0

can be viewedas defined on the space of sections of E′0 (still denoted by E′0), which is a contactinvariant. Since

(ΠE0)|E = ((ΠE)|E0

)−1, it follows that (ΠE′0)|E = ((ΠE)|E′0)−1

and dc viewed as an operator on E′0,

((ΠE)|E′0)−1 d (ΠE)|E′0is a contact invariant. In the sequel, we shall ignore the distinction between E0 andE′0. We shall denote by E•0 =

⊕hE

h0 endowed with the exterior differential dc.

Alternate contact invariant descriptions of Rumin’s complex can be found in [13]and [16].

By construction,

i) d2c = 0;

ii) the complex E0 := (E•0 , dc) is homotopically equivalent to the de Rhamcomplex Ω := (Ω•, d). Thus, if D ⊂ Hn is an open set, unambiguously wewrite Hh(D) for the h-th cohomology group;

iii) dc : Eh0 → Eh+10 is a homogeneous differential operator in the horizontal

derivatives of order 1 if h 6= n, whereas dc : En0 → En+10 is an homogeneous

differential operator in the horizontal derivatives of order 2.

The following statement expresses the fact that Rumin’s complex is invariantunder contactomorphism. In other words, the pull-back map is natural i.e. it is achain map for (E•0 , dc).

Proposition 2.14. If φ is a contactomorphism from an open set U ⊂ Hn to M ,and we denote by V the open set V := φ(U), the pull-back operator φ# satisfies:


i) φ#E•0 (V) = E•0 (U);ii) dcφ

# = φ#dc;iii) if ζ is a smooth function in M , then the differential operator in U ⊂ Hn

defined by v → φ#[dc, ζ](φ−1)#v is a differential operator of order zero ifv ∈ Eh0 (U), h 6= n and a differential operator of order 1 if v ∈ En0 (U).

Proof. Assertions i) and ii) follow straightforwardly since φ is a contact map. As-sertion iii) follows from Lemma 2.11, since, by definition,

φ#[dc, ζ](φ−1)#v = [dc, ζ φ]v.

3. Kernels and Laplacians

3.1. Kernels in Heisenberg groups. Following a classical notation ([55]), if U ⊂Hn is an open set, we denote by D(U) the space of smooth functions in U withcompact support, by D′(U) the space of distributions in U , and by E ′(U) the spaceof compactly supported distributions in U .

If f is a real function defined in Hn, we denote by vf the function definedby vf(p) := f(p−1), and, if T ∈ D′(Hn), then vT is the distribution defined by〈vT |φ〉 := 〈T |vφ〉 for any test function φ.

Following e.g. [21], p. 15, we can define a group convolution in Hn: if, forinstance, f ∈ D(Hn) and g ∈ L1

loc(Hn), we set

(20) f ∗ g(p) :=

∫f(q)g(q−1 · p) dq for q ∈ Hn.

We remind that, if (say) g is a smooth function and P is a left invariant differentialoperator, then

P (f ∗ g) = f ∗ Pg.We remind also that the convolution is again well defined when f, g ∈ D′(Hn),provided at least one of them has compact support. In this case the followingidentities hold

(21) 〈f ∗ g|φ〉 = 〈g|vf ∗ φ〉 and 〈f ∗ g|φ〉 = 〈f |φ ∗ vg〉

for any test function φ.As in [21], we also adopt the following multi-index notation for higher-order

derivatives. If I = (i1, . . . , i2n+1) is a multi–index, we set

(22) W I = W i11 · · ·W

i2n2n T i2n+1 .

By the Poincare–Birkhoff–Witt theorem, the differential operators W I form a basisfor the algebra of left invariant differential operators in Hn. Furthermore, we set|I| := i1 + · · ·+ i2n + i2n+1 the order of the differential operator W I , and d(I) :=i1 + · · ·+ i2n + 2i2n+1 its degree of homogeneity with respect to group dilations.

Suppose now f, g ∈ D′(Hn) with f compactly supported. Then, if ψ ∈ D(Hn),we have

〈(W If) ∗ g|ψ〉 = 〈W If |ψ ∗ vg〉 = (−1)|I|〈f |ψ ∗ (W I vg)〉

= (−1)|I|〈f ∗ vW I vg|ψ〉.(23)


Definition 3.1. Let ω ∈ D(Hn) be supported in the unit ball B(e, 1), and as-sume

∫ω(x) dx = 1. If ε > 0, we denote by ωε the Friedrichs mollifier ωε(x) :=

ε−Qω(δ1/εx).The procedure of regularization by convolution can be extended componentwise

to differential forms in L1loc(Hn, E•0 ), as follows: if α =

∑j αjξ

hj , we set:

ωε ∗ α :=∑j

(ωε ∗ αj)ξhj .

As above, denote by ∗ the group convolution in Hn. By [21] and (23), if u ∈L1

loc(Hn), the convolution uε := u ∗ ωε enjoys the same properties of the usualregularizing convolutions in Euclidean spaces.

Following [20], we remind now the notion of kernel of type µ.

Definition 3.2. A kernel of type µ is a homogeneous distribution of degree µ−Q(with respect to group dilations δr), that is smooth outside of the origin.

The convolution operator with a kernel of type µ is called an operator of type µ.

Proposition 3.3. Let K ∈ D′(Hn) be a kernel of type µ.

i) vK is again a kernel of type µ;ii) WK and KW are associated with kernels of type µ− 1 for any horizontal

derivative W ;iii) If µ > 0, then K ∈ L1

loc(Hn).

Theorem 3.4. Suppose 0 < α < Q, and let K be a kernel of type α. Then

i) if 1 < p < Q/α, and 1/q := 1/p− α/Q, then there exists C = C(p, α) > 0such that

‖u ∗K‖Lq(Hn) ≤ C‖u‖Lp(Hn)

for all u ∈ Lp(Hn).ii) If p ≥ Q/α and B,B′ ⊂ Hn are fixed balls with B ⊂ B′, then for any q ≥ p

there exists C = C(B,B′, p, q, α) > 0

‖u ∗K‖Lq(B′) ≤ C‖u‖Lp(B)

for all u ∈ Lp(Hn) with supp u ⊂ B.iii) If K is a kernel of type 0 and 1 < p <∞, then there exists C = C(p) > 0

such that

‖u ∗K‖Lp(Hn) ≤ C‖u‖Lp(Hn).

Proof. For statements i) and iii), we refer to [20], Propositions 1.11 and 1.9. Asfor ii), if p ≥ Q/α, we choose 1 < p < Q/α such that 1/p ≤ 1/q + α/Q. If we set1/q := 1/p− α/Q < 1/q, then

‖u∗K‖Lq(B′) ≤ CB′‖u ∗K‖Lq(B′) ≤ CB′‖u ∗K‖Lq(Hn)

≤ C ′(B′)‖u‖Lp(Hn) ≤ C ′(B,B′)‖u‖Lp(B).

Lemma 3.5. Suppose 0 < α < Q. If K is a kernel of type α and ψ ∈ D(Hn),ψ ≡ 1 in a neighborhood of the origin, then the statements i) and ii) of Theorem3.4 still hold if we replace K by (1− ψ)K.

Analogously, if K is a kernel of type 0 and ψ ∈ D(Hn), then statement iii) ofTheorem 3.4 still hold if we replace K by (ψ − 1)K.


Proof. As in [20], Proposition 1.11, we merely need notice that |(1 − ψ)K(x)| ≤Cψ|x|α−Q, so that (1− ψ)K ∈ LQ/(Q−α),∞(Hn), and therefore i) and ii) hold true.

Suppose now α = 0.Notice that (ψ−1)K ∈ L1,∞(Hn), and therefore also u→ ((ψ−1)K)∗u is Lp−Lp

continuous by Hausdorff-Young Theorem. This proves that iii) holds true.

Remark 3.6. By Theorem 3.4, Lemma 3.5 still holds if we replace (1 − ψ)K byψK.

The following estimate will be useful in the sequel.

Lemma 3.7. Let g be a a kernel of type µ > 0. Then, if f ∈ D(Hn) and R is anhomogeneous polynomial of degree ` ≥ 0 in the horizontal derivatives, we have

R(f ∗ g)(p) = O(|p|µ−Q−`) as p→∞.

In addition, let g be a smooth function in Hn \ 0 satisfying the logarithmicestimate

|g(p)| ≤ C(1 + | ln |p||),

and suppose its horizontal derivatives are kernels of type Q−1 with respect to groupdilations. Then, if f ∈ D(Hn) and R is an homogeneous polynomial of degree ` ≥ 0in the horizontal derivatives, we have

R(f ∗ g)(p) = O(|p|−`) as p→∞ if ` > 0;

R(f ∗ g)(p) = O(ln |p|) as p→∞ if ` = 0.

Proof. The first part of the lemma is a particular instance of Lemma 6.4 in [21]. Asfor the second part, we can repeat the same argument. Indeed, the first statementfollows straightforwardly from the first part of the lemma, since, by the Poincare–Birkhoff–Witt theorem, we can write

R(f ∗ g) =∑j

R′`(f ∗Wjg) ,

where the differential operators R′j have homogeneous degree ` − 1. Finally, thelast statement can be proved as follows: suppose supp f ⊂ B(0,M), M > 1, andtake |p| > 2M . Then, keeping in mind that 1 < M ≤ |q−1p| ≤M + |p|,∣∣R(f∗g)(p)

∣∣ ≤ ∫ |f(q)| |g(q−1p)| dq ≤ C∫|f(q)| (1 + ln |q−1p|) dq

≤ C∫|f(q)| (1 + ln(M + |p|)) dq ≤ C ′(1 + ln |p|).

3.2. Rumin’s Laplacians. In this section we recall the main properties of theRumin’s generalization of Laplace operator in Heisenberg groups. In order tointroduce this operator we need preliminarily the following property about the L2-adjoint of Rumin’s exterior differential dc.

Proposition 3.8. Denote by δc = d∗c the formal adjoint of dc in L2(Hn, E∗0 ). Thenδc = (−1)h ∗ dc∗ on Eh0 .


Definition 3.9. In Hn, following [50], we define the operator ∆H,h on Eh0 by setting

∆H,h =

dcδc + δcdc if h 6= n, n+ 1;(dcδc)

2 + δcdc if h = n;dcδc + (δcdc)

2 if h = n+ 1.

For the sake of simplicity, since a basis of Eh0 is fixed, any α ∈ Eh0 can beidentified with the vector (α1, . . . , αNh

) of its components, and the operator ∆H,hcan be identified with a matrix-valued map, still denoted by ∆H,h

(24) ∆H,h = (∆ijH,h)i,j=1,...,Nh

: D′(Hn,RNh)→ D′(Hn,RNh),

whereNh is the dimension of Eh0 (Nh is explicit in Theorem 2.5, iv)) andD′(Hn,RNh)is the space of vector-valued distributions on Hn .

This identification makes possible to avoid the notion of currents: we refer to [8]for a more elegant presentation.

Remark 3.10. We stress that ∆H,h is a left invariant differential operator of order2 if h 6= n, n+ 1 and of order 4 if h = n, n+ 1 with respect to group dilations (see

(9)), i.e. its components ∆ijH,h can be written, with the notations of (22), as

∆ijH,h =

∑I

cijI WI

with d(I) = 2 if h 6= n, n + 1 and d(I) = 4 if h = n, n + 1. In addition, −∆H,0 =∑2nj=1(W 2

j ) is the usual sub-Laplacian of Hn.

Remark 3.11. As a straightforward consequence of Proposition 3.8 for any 0 ≤h ≤ 2n+ 1 we have ∗∆H,h = ∆H,2n+1−h∗ (since (2n+ 1) is odd).

It is proved in [50] that ∆H,h is hypoelliptic and maximal hypoelliptic in thesense of [32]. In general, if L is a differential operator on D′(Hn,RNh), then Lis said hypoelliptic if for any open set V ⊂ Hn where Lα is smooth, then α issmooth in V. In addition, if L is homogeneous of degree a ∈ N, we say that L ismaximal hypoelliptic if for any δ > 0 there exists C = C(δ) > 0 such that for anyhomogeneous polynomial P in W1, . . . ,W2n of degree a we have

‖Pα‖L2(Hn,RNh ) ≤ C(‖Lα‖L2(Hn,RNh ) + ‖α‖L2(Hn,RNh )

)for any α ∈ D(B(0, δ),RNh).

The next theorem provides a key tool for the present paper: the existence of asuitable “inverse” ∆−1

H,h of ∆H,h that is associated with a vector-valued kernel, that

we still denote by ∆−1H,h.

Combining [50], Section 3, and [10], Theorems 3.1 and 4.1, we obtain the follow-ing result. We stress again the fact that the order of ∆H,h with respect to groupdilations is 2 if h 6= n, n+ 1 whereas 4 if h = n, n+ 1.

Theorem 3.12 (see [3], Theorem 4.6). If 0 ≤ h ≤ 2n + 1, denote by a the orderof ∆H,h with respect to group dilations (by Remark 3.10, a = 2 if h 6= n, n+ 1 anda = 4 if h = n, n+ 1). Then there exist

(25) Kij ∈ D′(Hn) ∩ C∞(Hn \ 0) for i, j = 1, . . . , Nh,

with the following properties:


i) if a < Q then the Kij’s are kernels of type a, for i, j = 1, . . . , Nh. Ifa = Q, then the Kij’s satisfy the logarithmic estimate |Kij(p)| ≤ C(1 +| ln ρ(p)|) and hence belong to L1

loc(Hn). Moreover, their horizontal deriva-tives W`Kij, ` = 1, . . . , 2n, are kernels of type Q− 1;

ii) when α = (α1, . . . , αNh) ∈ D(Hn,RNh) (here again Nh = dimEh0 ), if we

set

(26) ∆−1H,hα :=

(∑j

αj ∗K1j , . . . ,∑j

αj ∗KNhj

),

then

∆H,h∆−1H,hα = α.

Moreover, if a < Q, also

∆−1H,h∆H,hα = α.

iii) if a = Q, then for any α ∈ D(Hn,RNh) there exists βα := (β1, . . . , βNh) ∈

RNh , such that

∆−1H,h∆H,hα− α = βα.

Remark 3.13. Coherently with formula (24), the matrix-valued operator ∆−1H,h can

be identified with an operator (still denoted by ∆−1H,h) acting on smooth compactly

supported differential forms in D(Hn, Eh0 ). Moreover, when the notation will not bemisleading, we shall denote by ∆−1

H,h its kernel.

The following lemma states that dc and ∆H commute.

Lemma 3.14. We notice that the Laplace operator commutes with the exteriordifferential dc. More precisely, if α ∈ C∞(Hn, Eh0 ) and n ≥ 1,

i) dc∆H,hα = ∆H,h+1dcα, h = 0, 1, . . . , 2n, h 6= n− 1, n.ii) dcδcdc∆H,n−1α = ∆H,ndcα, (h = n− 1).iii) dc∆H,nα = dcδc∆H,n+1dcα (h = n).iv) dcδc∆H,nα = ∆H,ndcδcα (h = n).

Proof. The proof is an easy consequence of the fact that d2c = 0. Indeed, let us prove

i). Since h 6= n − 1, n, we write dc∆H,hα = dc(dcδc + δcdc)α = d2cδcα + dcδcdcα =

dcδcdcα = dcδcdcα+ δcd2cα = (dcδc + δcdc)dcα = ∆H,h+1dcα.

To prove ii), we write dcδcdc∆H,n−1α = dcδcdc(dcδc + δcdc)α = dcδcd2cδcα +

dcδcdcδcdcα = (dcδc)2dcα = (dcδc)

2dcα + (δcdc)dcα = ∆H,ndcα. An analogousargument applies to iii) and iv).

The commutation of dc and δc with ∆−1H follows from the previous lemma:

Lemma 3.15. If α ∈ D(Hn, Eh0 ) and n ≥ 1,

i) dc∆−1H,hα = ∆−1

H,h+1dcα, h = 0, 1, . . . , 2n, h 6= n− 1, n+ 1.

ii) dc∆−1H,n−1α = dcδc∆

−1H,ndcα (h = n− 1).

iii) dcδcdc∆−1H,n+1α = ∆−1

H,n+2dcα, (h = n+ 1).

iv) δc∆−1H,hα = ∆−1

H,h−1δcα h = 1, . . . , 2n+ 1, h 6= n, n+ 2.

v) δc∆−1H,n+2α = δcdc∆

−1H,n+1δcα (h = n+ 2).

vi) δcdcδc∆−1H,nα = ∆−1

H,n−1δcα, (h = n).


Proof. Let us prove i), ii), iii). Put

ωh : = dc∆−1H,hα−∆−1

H,h+1dcα if h 6= n− 1, n+ 1,

ωn−1 : = dc∆−1H,n−1α− dcδc∆

−1H,ndcα

ωn+1 : = dcδcdc∆−1H,n+1α−∆−1

H,n+2dcα.

We notice first that, by Theorem 3.12 and Proposition 3.3, for all h = 1, . . . , 2n,ωh = Mh ∗ α, where Mh is a kernel of type 1. Thus, by Lemma 3.7

(27) ωh(x) = O(|x|1−Q) as x→∞.

We claim that

(28) ∆H,h+1ωh = 0 for h = 1, . . . , 2n.

Thus, by [10], Proposition 3.2, ωh is a form with polynomial coefficients. Then, by(27) necessarily ωh ≡ 0. Thus we have proved i), ii), iii).

We are left then with the proof of the claim (28).Suppose first h 6= n − 1, n, n + 1. By Lemma 3.14-i) and by Theorem 3.12, we

have:

∆H,h+1ωh = ∆H,h+1dc∆−1H,hα−∆H,h+1∆−1

H,h+1dcα

= dc∆H,h∆−1H,hα− dcα = 0.

If h = n− 1 then, by Lemma 3.14-iii),iv) and by Theorem 3.12

∆H,nωn−1 = ∆H,n

(dc∆

−1H,n−1α− dcδc∆

−1H,ndcα

)= dcδcdc∆H,n−1∆−1

H,n−1α− dcδc∆H,n∆−1H,ndcα = 0.

If h = n, then (keeping in mind that dc∆−1H,nα is a form of degree n + 1 and

∆−1H,nα is a form of degree n, we use again Lemma 3.14-i))

∆H,n+1ωn = ∆H,n+1(dc∆−1H,nα−∆−1

H,n+1dcα)

= dc∆H,n∆−1H,nα− dcα = 0.

Finally, if h = n+ 1 then, again Lemma 3.14-i),

∆H,n+2ωn+1 = ∆H,n+2(dc∆−1H,n+1α−∆−1

H,n+2dcα)

= dc∆H,n+1∆−1H,n+1α− dcα = 0.

This proves (28) and hence we have proved i), ii), iii).Since δc = (−1)h ∗ dc∗ , and keeping in mind Remark 3.11, the remaining asser-

tions iv), v), vi) follow by the Hodge duality from i), ii), iii).

4. Function spaces

4.1. Sobolev spaces on Heisenberg groups. Let U ⊂ Hn be an open set andlet 1 ≤ p ≤ ∞ and m ∈ N, Wm,p

Euc (U) denotes the usual Sobolev space. We wantnow to introduce intrinsic (horizontal) Sobolev spaces.

Since here we are dealing only with integer order Folland-Stein function spaces,we can give this simpler definition (for a general presentation, see e.g. [20]).


Definition 4.1. If U ⊂ Hn is an open set, 1 ≤ p ≤ ∞ and m ∈ N, then the spaceWm,p(U) is the space of all u ∈ Lp(U) such that, with the notation of (22),

W Iu ∈ Lp(U) for all multi-indices I with d(I) ≤ m,

endowed with the natural norm that we denote by

‖u‖Wk,p(U) :=∑

d(I)≤m

‖W Iu‖Lp(U).

Folland-Stein Sobolev spaces enjoy the following properties akin to those of theusual Euclidean Sobolev spaces (see [20], and, e.g. [25]).

Theorem 4.2. If U ⊂ Hn, 1 ≤ p <∞, and k ∈ N, then

i) W k,p(U) is a Banach space;ii) W k,p(U) ∩ C∞(U) is dense in W k,p(U);

iii) if U = Hn, then D(Hn) is dense in W k,p(U).

Definition 4.3. If U ⊂ Hn is open and if 1 ≤ p < ∞, we denote byW k,p(U) the

completion of D(U) in W k,p(U).

Remark 4.4. If U ⊂ Hn is bounded, by (iterated) Poincare inequality (see e.g.[34]), it follows that the norms

‖u‖Wk,p(U) and∑

d(I)=k

‖W Iu‖Lp(U)

are equivalent onW k,p(U) when 1 ≤ p <∞.

If U ⊂ Hn is an open set and 1 < p < ∞, W−k,p(U) is the dual space ofW k,p′(U), where 1/p+ 1/p′ = 1.

Remark 4.5. It is well known that

W−k,p(U) = f0+∑

d(I)=k

W IfI : f0, fI ∈ Lp(U) for any multi-index I such that d(I) = k,

and

‖u‖W−k,p(U) ≈ inf‖f0‖Lp(U) +∑I

‖fI‖Lp(U) : d(I) = k, f0 +∑

d(I)=k

W IfI = u.

If U is bounded, then we can take f0 = 0.Finally, we stress that

f0 +∑

d(I)=k

W IfI , f0, fI ∈ D(U) for any I multi-index such that d(I) = k

is dense in W−k,p(U).

Definition 4.6. If U ⊂ Hn is an open set, 0 ≤ h ≤ 2n+ 1, 1 ≤ p ≤ ∞ and m ≥ 0,

we denote by Wm,p(U,∧h

h) (byWm,p(U,

∧hh)) the space of all sections of

∧hh

such that their components with respect to a given left-invariant frame belong to

Wm,p(U) (toWm,p(U), respectively), endowed with its natural norm.


The spaces Wm,p(U,Eh0 ) andWm,p(U,Eh0 ) are defined in the same way. In

addition,

W−m,p(U,Eh0 ) :=( W

m,p′(U,Eh0 ))∗.

Clearly, these definitions are independent of the choice of the frame itself.On the other hand, the spaces W−m,p(U,Eh0 ) can be viewed as spaces of currents

on (E•0 , dc) as in [8], Proposition 3.14. More precisely we have:

Remark 4.7. As in [8], Proposition 3.14, an element of W−m,p(U,Eh0 ) can beidentified (with respect to our basis) with a Nh-tuple

(T1, . . . , TNh) ∈

(W−m,p(U,Eh0 )

)Nh

.

This is nothing but the intuitive notion of “currents as differential forms with distri-butional coefficients”. The action of u ∈W−m,p(U,Eh0 ) associated with (T1, . . . , TNh

)

on the form∑j αjξ

hj ∈

Wm,p′(U,Eh0 ) is given by

〈u|α〉 :=∑j

〈Tj |αj〉.

On the other hand, suppose for the sake of simplicity that U is bounded, then byRemark 4.5 there exist f jI ∈ Lp(U), j = 1, . . . , Nh, i = 1, . . . , 2n+ 1 such that

(29) 〈u|α〉 =∑j

∑d(I)=m

∫U

f jI (x)W Iαj(x) dx.

Alternatively, one can express duality in spaces of differential forms using thepairing between h-forms and (2n+ 1− h)-forms defined by

(α, β) 7→∫U

α ∧ β.

Note that this makes sense for Rumin forms and is a nondegenerate pairing. In thismanner, the dual of Lp(U,Eh0 ) is Lp

′(U,E2n+1−h

0 ). Hence W−m,p(U,Eh0 ) consists ofdifferential forms of degree 2n+ 1−h whose coefficients are distributions belongingto W−m,p(U).

In the Riemannian setting, Sobolev spaces of differential forms are invariantwith respect to the pull-back operator associated with sufficiently smooth diffeo-morphisms (see, e.g. [55], Lemma 1.3.9). An analogous statement holds for Folland-Stein Sobolev spaces in Heisenberg groups, provided we restrict ourselves to contactdiffeomorphisms. Indeed we have:

Lemma 4.8. If k is a positive integer, let U , V be open subsets of Hn. Let φ :U → V be a Ck-bounded contact diffeomorphism. Let ` = −k + 1, . . . , k − 1. Thenthe pull-back operator φ] from W `,p forms on V to W `,p forms on U is bounded,and its norm depends only on the Ck norms of φ and φ−1.

Proof. When ` ≥ 0, this follows from the chain rule and the change of variablesformula. According to the change of variables formula∫

U

φ]α ∧ φ]β =

∫V

α ∧ β,

the adjoint of φ] with respect to the above pairing is (φ−1)]. Hence φ] is boundedon negative Sobolev spaces of differential forms as well.


4.2. Sobolev spaces on contact sub-Riemannian manifolds with boundedgeometry. First of all, let us give the definition of contact manifolds of boundedgeometry.

Definition 4.9. Let k be a positive integer and let B(e, 1) denote the unit sub-Riemannian ball in Hn. We say that a sub-Riemannian contact manifold (M,H, g)has bounded Ck-geometry is there exist constants r, C > 0 such that, for everyx ∈ M , there exists a contactomorphism (i.e. a diffeomorphism preserving thecontact forms) φx : B(e, 1)→M that satisfies

(1) B(x, r) ⊂ φx(B(e, 1));(2) φx is C-bi-Lipschitz, i.e.

(30)1

Cd(p, q) ≤ dM (φx(p), φx(q)) ≤ Cd(p, q) for all p, q ∈ B(e, 1);

(3) coordinate changes φx φ−1y and their first k derivatives with respect to unit

left-invariant horizontal vector fields are bounded by C.

Remark 4.10. Compact sub-Riemannian contact manifolds have bounded geom-etry. More examples arise from covering spaces of such compact manifolds. Notethat every orientable compact 3-manifold admits a contact structure ([39]), it canbe equipped with sub-Riemannian structures, its universal covering space is usu-ally noncompact. This leads to a large variety of non-compact bounded geometrysub-Riemannian contact 3-manifolds.

The following covering lemma is basically [40], Theorem 1.2.

Lemma 4.11. Let (M,H, g) be a bounded Ck-geometry sub-Riemannian contactmanifold, where k is a positive integer. Then there exists ρ > 0 (depending only onthe radius r of Definition 4.9) and an at most countable covering B(xj , ρ) of Msuch that

i) each ball B(xj , ρ) is contained in the image of one of the contact charts ofDefinition 4.9;

ii) B(xj ,15ρ) ∩B(xi,

15ρ) = ∅ if i 6= j;

iii) the covering is uniformly locally finite. Even more, there exists a N =N(M) ∈ N such that for each ball B(x, ρ)

#k ∈ N such that B(xk, ρ) ∩B(x, ρ) 6= ∅ ≤ N.In addition, if B(xk, ρ) ∩ B(x, ρ) 6= ∅, then B(xk, ρ) ⊂ B(x, r), whereB(x, r) has been defined in Definition 4.9-(2)).

Proof. First we notice that M is separable. Indeed, let x ∈ M be fixed. With thenotations of Definition 4.9, if we set φx(B(0, 1)) := Ux then Ux, x ∈M is an opencovering of M . Let now Vxj , j ∈ N be a countable refinement of Ux, x ∈M (see

[58], Lemma 1.9). For any j ∈ N, let Sj be a countable dense subset of φ−1xj

(Vxj);

then φxj(Sj) is a countable dense subset of Vxj

. Thus Σ := ∪jφxj(Sj) is a countable

dense subset of M .Let now ρ ∈ (0, r/2) be fixed. Then, by [40], Theorem 1,2, there exists a family

of disjoint balls B(xα,ρ5 ) such that B(xα, ρ) is an open covering of M . We

prove now that we can extract a countable sub-family B(xαj , ρ) =: B(xj , ρ)which is still an open covering of M . Indeed, for any y ∈ Σ, let us prove that#α such that y ∈ B(xα, ρ) ≤ N , where N is a geometric constant. If y ∈B(xα, ρ)∩B(xβ , ρ), then dM (xα, xβ) < 2ρ. In addition, then B(xα, ρ) and B(xβ , ρ)


are contained in φy(B(e, 1)) since 2ρ < r. From now on we assume ρ > 0 is fixedwith 3ρ < r. We notice that by (30)

B(φ−1y (xα), ρ/5C) ⊂ φ−1

y (B(xα, ρ/5)).

so that, if α, β ∈ α such that y ∈ B(xα, ρ), then

B(φ−1y (xα), ρ/5C) ∩B(φ−1

y (xβ), ρ/5C) = ∅.

By the doubling property of the Lebesgue measure in Hn with respect to Cygan-Koranyi’s balls, this is possible only for at most N balls B(φ−1

y (xα), ρ/5C) whereN = N(C, ρ). It follows that B(xα, ρ), such that y ∈ B(xα, ρ) and y ∈ Σ is acountable subfamily of balls B(xj , ρ), j ∈ N ⊂ B(xα, ρ) such that the ballsB(xj , ρ/5), j ∈ N are disjoint. Finally, we notice that, by the density of Σstraightforwardly B(xj , ρ), j ∈ N still covers all M , since the balls have the sameradius ρ.

Finally, notice that our previous arguments yield also that the covering is uni-formly locally finite. Indeed, let x be fixed and let B(xk, ρ) ∩ B(x, ρ) 6= ∅. ThenB(xk, ρ) ⊂ B(x, 3ρ) ⊂ B(x, r) since 3ρ < r. Consider now the family of open sets

B := φ−1x (B(xk, ρ/5)), with B(xk, ρ) ∩B(x, ρ) 6= ∅.

By definition, the open sets of B are disjoint (by ii)), and their union is containedin B(e, 1). In addition, again by (30)

B(φ−1x (xk), ρ/5C) ⊂ φ−1

x (B(xk, ρ/5)),

and the assertion follows again by a doubling argument in Hn.

We can define now Sobolev spaces (involving a positive or negative number ofderivatives) on bounded geometry contact sub-Riemannian manifolds.

Definition 4.12. Let k be a positive integer, and let (M,H, g) be a bounded Ck-geometry sub-Riemannian contact manifold, and let χj be a partition of unitysubordinate to the atlas U := B(xj , ρ), φxj of Lemma 4.11. From now on, for

sake of simplicity, we shall write φj := φxj. We stress that φ−1

j (supp χj) ⊂ B(e, 1).

If α is a Rumin differential form on M , we say that α ∈ W `,pU (M,E•0 ) for ` ∈ Z,

−k + 1 ≤ ` ≤ k − 1 and p ≥ 1 if

φ#j (χjα) ∈W `,p(Hn, E•0 ) for j ∈ N

(notice that φ#j (χjα) is compactly supported in B(e, 1) and therefore can be contin-

ued by zero on all of Hn). Then we set

‖α‖W `,pU (M,E•0 ) :=

∑j

‖φ#j (χjα)‖p

W `,p(Hn,E•0 )

1/p

.

The following result shows that the definition of the Sobolev spaces W `,pU (M,E•0 )

does not depend on the atlas U . Therefore, once the proposition is proved, we dropthe index U from the notation for Sobolev norms.

Proposition 4.13. Let k and ` be as above, and let (M,H, g) be a bounded Ck-geometry sub-Riemannian contact manifold. If U ′ := B(yj , ρ

′), φ′yj is another


atlas of M satisfying Definition 4.9 and Lemma 4.11 with the same choice of ρ,and χ′j is an associated partition of unity, then

W `,pU (M,E•0 ) = W `,p

U ′ (M,E•0 ),

with equivalent norms.

Proof. Let j ∈ N be fixed, and let (B(xj , ρ), φj) be a chart of U . We can write

χj =∑k∈Ij

χ′kχj ,

where #Ij ≤ N , since, by Lemma 4.11 iii), B(xj , ρ) is covered by at most N ballsof the covering associated with U ′. Thus, by Definition 4.9-(3) and keeping in mindthat suppχ′k ⊂ B(xj , r) (since 3ρ < r), we have

‖φ#j (χjα)‖W `,p(Hn,E•0 ) ≤

∑k∈Ij

‖φ#j (χ′kχjα)‖W `,p(Hn,E•0 )

≤ c∑k∈Ij

‖φ#j (χ′kα)‖W `,p(Hn,E•0 )

= c∑k∈Ij

‖(φjφ′−1k )#φ′#k (χ′kα)‖W `,p(Hn,E•0 )

≤ c∑k∈Ij

‖φ′#k (χ′kα)‖W `,p(Hn,E•0 )

≤ cN‖α‖W `,p

U′ (M,E•0 ).

5. Homotopy formulae and Poincare and Sobolev inequalities

In this paper we are mainly interested to obtain functional inequalities for differ-ential forms that are the counterparts of the classical (p, q)-Sobolev and Poincareinequalities on a ball B ⊂ Rn with sharp exponents of the form

‖u− uB‖Lq (B) ≤ C(r)‖∇u‖Lp(B)

(as well as of its counterpart for compactly supported functions). In this case, wecan choose q = pn/(n− p), provided p < n.

Definition 5.1. Take λ > 1 and set B = B(e, 1) and B′ = B(e, λ). If 1 ≤ h ≤2n + 1 and q ≥ p ≥ 1, we say that the interior H-Poincarep,q inequality holds inEh0 if there exists a constant C such that, for every dc-exact differential k-form ω

in Lp(B′;Eh0 ) there exists a differential (k − 1)-form φ in Lq(B,Eh−10 ) such that

dcφ = ω and

‖φ‖Lq(B,Eh−10 ) ≤ C ‖ω‖Lp(B′,Eh

0 ) interiorH-Poincarep,q(h).

Remark 5.2. As we pointed out in Section 1.1, what we call interiorH-Poincarep,q(h)is a slightly weaker formulation of the standard Poincare inequality where B = B′.The word “interior” is meant to stress that the inequality is not affected by the ge-ometry of the boundary of the ball. This is obtained thanks to a “loss on domain”,passing from a larger ball B′ to a smaller ball B.


Remark 5.3. If h = 1 and Q > p ≥ 1, then (H-Poincarep,q(1)) is nothing but the

usual Poincare inequality with1

p− 1

q=

1

Q(see e.g. [23], [17], [38]).

Remark 5.4. If we replace Rumin’s complex (E•0 , dc) by the usual de Rham’s com-plex (Ω•, d) in R2n+1, then the H-Poincarep,q inequality holds on Euclidean balls forh = 1 and n > p ≥ 1. If h > 1, then the H-Poincarep,q inequality for 2n+1 > p > 1

and1

p− 1

q=

1

2n+ 1is proved by Iwaniec & Lutoborski (see [33], Corollary 4.2).

We give below a statement that deals with H-Sobolev inequality.

Definition 5.5. Take λ > 1 and set B = B(e, 1) and B′ = B(e, λ). If 1 ≤ h ≤ 2n,1 ≤ p ≤ q <∞ and q ≥ p, we say that the interior H-Sobolevp,q(h) inequality holdsif there exists a constant C such that for every compactly supported smooth dc-exact differential h-form ω in Lp(B;Eh0 ) there exists a smooth compactly supported

differential (h− 1)-form φ in Lq(B′, Eh−10 ) such that dcφ = ω in B′ and

‖φ‖Lq(B′,Eh−10 ) ≤ C ‖ω‖Lp(B,Eh

0 ).(31)

Here we have extended ω by 0 to all of B′.

Remark 5.6. If h = 1 and Q > p ≥ 1, then (H-Sobolevp,q(1)) is nothing but the

usual Sobolev inequality with1

p− 1

q=

1

Q.

In [33], starting from Cartan’s homotopy formula, the authors proved that, ifD ⊂ RN is a convex set, 1 < p <∞, 1 < h < N , then there exists a bounded linearmap:

(32) KEuc,h : Lp(D,∧

h)→W 1,p(D,∧

h−1)

that is a homotopy operator, i.e.

(33) ω = dKEuc,hω +KEuc,h+1dω for all ω ∈ C∞(D,∧h)

(see Proposition 4.1 and Lemma 4.2 in [33]). More precisely, KEuc has the form

(34) KEuc,hω(x) =

∫D

ψ(y)Kyω(x) dy,

where ψ ∈ D(D),∫Dψ(y) dy = 1, and

〈Kyω(x)|ξ1 ∧ · · · ∧ ξh−1)〉 :=

∫ 1

0

th−1〈ω(tx+ (1− t)y)|(x− y) ∧ ξ1 ∧ · · · ∧ ξh−1)〉.

(35)

Starting from [33], in [42], Theorem 4.1, the authors define a compact homo-topy operator JEuc,h in Lipschitz star-shaped domains in the Euclidean space RN ,providing an explicit representation formula for JEuc,h, together with continuityproperties among Sobolev spaces. More precisely, if D ⊂ RN is a star-shapedLipschitz domain and 1 < h < N , then there exists

JEuc,h : Lp(D,∧

h)→W 1,p0 (D,

∧h−1)

such that

ω = dJEuc,hω + JEuc,h+1dω for all ω ∈ D(D,∧h).


Furthermore, JEuc,h maps smooth compactly supported forms to smooth compactlysupported forms.

Take now D = B(e, 1) =: B and N = 2n+ 1. If ω ∈ C∞(B,Eh0 ), then we set

K = ΠE0ΠE KEuc ΠE(36)

(for the sake of simplicity, from now on we drop the index k - the degree of theform - writing, e.g., KEuc instead of KEuc,hk).

Analogously, we can define

J = ΠE0 ΠE JEuc ΠE .(37)

Then K and J invert Rumin’s differential dc on closed forms of the same degree.More precisely, we have:

Lemma 5.7. If ω is a smooth dc-exact differential form, then

(38) ω = dcKω if 1 ≤ h ≤ 2n+ 1 and ω = dcJω if 1 ≤ h ≤ 2n+ 1.

In addition, if ω is compactly supported in B, then Jω is still compactly supportedin B.

Proof. We prove for instance the identity for dcKω. If dcω = 0, then d(ΠEω) = 0,and hence

ΠEω = dKEuc(ΠEω),

by (33). We recall now that , by Theorem 2.7 ii) and iv), dΠE = ΠEd and bothΠEΠE0

ΠE = ΠE and ΠE0ΠEΠE0

= ΠE0. Thus, by (36),

dcKω = ΠE0dΠEΠE0ΠEKEucΠEω = ΠE0dΠEKEucΠEω

= ΠE0ΠEdKEucΠEω = ΠE0ΠEΠEω = ΠE0ΠEω

= ΠE0ΠEΠE0ω = ΠE0ω = ω,

since ω ∈ E•0 . Finally, if suppω ⊂ B, then supp Jω ⊂ B since both ΠE and ΠE0

preserve the support.

Lemma 5.8. Put B = B(e, 1). Then:

i) if 1 < p <∞ and h = 1, . . . , 2n+ 1, then K : W 1,p(B,Eh0 )→ Lp(B,Eh−10 )

is bounded;ii) if 1 < p <∞ and n+ 1 < h ≤ 2n+ 1, then K : Lp(B,Eh0 )→ Lp(B,Eh−1

0 )is compact;

iii) if 1 < p < ∞ and h = n + 1, then K : Lp(B,En+10 ) → Lp(B,En0 ) is

bounded.

Analogous assertions hold for 1 ≤ h ≤ 2n+1 when we replace K by J . In addition,supp Jω ⊂ B.

Proof. By its very definition, ΠE : W 1,p(B,Eh0 )→ Lp(B,Eh0 ) is bounded. By (32),

KEuc is continuous from Lp(B,Eh0 ) to W 1,p(B,Eh−10 ). Then we can conclude the

proof of i), keeping again into account that ΠE is a differential operator of order≤ 1 in the horizontal derivatives.

To prove ii) it is enough to remind that K = ΠE0KEuc on forms of degree h > n,

together with Remark 4.1 in [33].As for iii), the statement can be proved similarly to i), noticing that K =

ΠE0ΠEKEuc on forms of degree n+ 1.

Finally, supp Jω ⊂ B since both ΠE and ΠE0preserve the support.


The operators K and J provide a local homotopy in Rumin’s complex, but failto yield the Sobolev and Poincare inequalities we are looking for, since, becauseof the presence of the projection operator ΠE (that on forms of low degree is afirst order differential operator) they loose regularity as is stated in Lemma 5.8, ii)above. In order to build “good” local homotopy operators with the desired gain ofregularity, we have to combine them with homotopy operators which, though notlocal, in fact provide the “good” gain of regularity.

Proposition 5.9. If α ∈ D(Hn, Eh0 ) for p > 1 and h = 1, . . . , 2n, then the follow-

ing homotopy formulas hold: there exist operators K1, K1 and K2, K1 acting onD(Hn, E•0 ) such that

• if h 6= n, n+ 1, then α = dcK1α+ K1dcα, where K1 and K1 are associatedwith kernels k1, k1 of type 1;• if h = n, then α = dcK1α + K2dcα, where K1 and K2 are associated with

kernels k1, k2 of type 1 and 2, respectively;• if h = n + 1, then α = dcK2α + K1dcα, where K2 and K1 are associated

with kernels k2, k1 of type 2 and 1, respectively.

Proof. Suppose h 6= n− 1, n, n+ 1. By Lemma 3.15, we have:

α = ∆H,h∆−1H,hα = dc(δc∆

−1H,h)α+ δc(dc∆

−1H,h)α

= dc(δc∆−1H,h)α+ (δc∆

−1H,h+1)dcα.

where δc∆−1H,h and δc∆

−1H,h+1 are associated with a kernel of type 1 (by Proposition

3.3 and Theorem 3.12).Analogously, if h = n− 1

α = ∆H,n−1∆−1H,n−1α = dc(δc∆

−1H,n−1)α+ δc(dc∆

−1H,n−1)α

= dc(δc∆−1H,n−1)α+ (δcdcδc∆

−1H,n)dcα.

Again δc∆−1H,n−1 and δcdcδc∆

−1H,n are associated with kernels of type 1.

Take now h = n. Then

α = ∆H,n∆−1H,nα = (dcδc)

2∆−1H,nα+ δc(dc∆

−1H,n)α

= dc(δcdcδc∆−1H,n)α+ δc∆

−1H,n+1dcα

where δcdcδc∆−1H,n and δc∆

−1H,n+1 are associated with a kernel of type 1 and 2, re-

spectively).Finally, take h = n+ 1. Then

α = ∆H,n+1∆−1H,n+1α = dcδc∆

−1H,n+1α+ (δcdc)

2∆−1H,n+1α

= dcδc∆−1H,n+1α+ δc∆

−1H,n+2dcα

where δc∆−1H,n+1 and δc∆

−1H,n+2 associated with kernels of type 2 and 1, respectively.

The Lp −Lq continuity properties of convolution operators associated with Fol-land’s kernels yields the following global H-Poincarep,q(h) inequality in Hn (theglobal H-Sobolevp,q(h) is obtained in Corollary 5.21).


Corollary 5.10. Take 1 ≤ h ≤ 2n + 1. Suppose 1 < p < Q if h 6= n + 1 and1 < p < Q/2 if h = n+ 1. Let q ≥ p defined by

1

p− 1

q:=

1Q if h 6= n+ 1,2Q if h = n+ 1.

(39)

Then for any exact form α ∈ D(Hn, Eh0 ) there exists φ ∈ Lq(Hn, Eh−10 ) such that

dcφ = α and

‖φ‖Lq(Hn,Eh−10 ) ≤ C‖α‖Lp(Hn,Eh−1

0 )

(i.e., the global H-Poincarep,q(h) inequality holds for 1 ≤ h ≤ 2n+ 1).

Example 5.11. Suppose for instance n = 1. In this case Q = 4 and, keeping inmind Example 2.8, α = α1dx + α2dy ∈ E1

0 , then Corollary 5.10 yields that thereexists a function φ such that

Xφ = α1, Y φ = α2.

Moreover, if α = α13dx∧θ+α23dy∧θ ∈ E20 , then there exists φ = φ1dx+φ2dy ∈ E1

0 ,such that

X2φ2 − 2XY φ1 + Y Xφ1 = α13 and 2Y Xφ2 − Y 2φ1 −XY φ2 = α23.

Theorem 5.12. Let B = B(e, 1) and B′ = B(e, λ), λ > 1, be concentric balls

of Hn. If 1 ≤ h ≤ 2n + 1, there exist operators T and T from C∞(B′, Eh0 ) to

C∞(B,Eh−10 ) and S from C∞(B′, Eh0 ) to C∞(B,Eh0 ) satisfying

(40) dcT + T dc + S = I on B.

Proof. Suppose first h 6= n, n + 1. We consider a cut-off function ψR supportedin a R-neighborhood of the origin, such that ψR ≡ 1 near the origin. With thenotations of Proposition 5.9, we can write

(41) k1 = k1ψR + (1− ψR)k1 and k1 = k1ψR + (1− ψR)k1,

where

(42) k1 =: (k1)`,λ and k1 =: (k1)`,λ

are the matrix-valued kernels associated with the operators δc∆H,h and δc∆H,h+1,respectively, as shown in the proof of Proposition 5.9.

Let us denote by K1,R, K1,R the convolution operators associated with ψRk1,

ψRk1, respectively. Let us fix two balls B0, B1 with

(43) B b B0 b B1 b B′,

and a cut-off function χ ∈ D(B1), χ ≡ 1 on B0. If α ∈ C∞(B′, E•0 ), we set α0 = χα,continued by zero outside B1.

We have:

(44) α0 = dcK1,Rα0 + K1,Rdcα0 + S0α0,

where S0 is

(45) S0α0 := dc(α0 ∗ (1− ψR)k1) + dcα0 ∗ (1− ψR)k1.

We set

(46) Tα := K1,Rα0, T dcα := K1,Rdcα0, Sα := S0α0.


We notice that, provided R > 0 is small enough, the definition of T and T doesnot depend on the continuation of α outside B0. By (44) we have

α = dcTα+ T dcα+ Sα in B.

If h = n we can carry out the same construction, replacing k1 by k2 (keep in mind

that k2 is a kernel of type 2, again by Proposition 5.9). Analogously, if h = n + 1we can carry out the same construction, replacing k1 by k2 (again a kernel of type2).

Later on, we need the following remark:

Remark 5.13. By construction, if suppα ⊂ B then suppTα is contained in aR-neighborhood of B and then is contained in B0 provided R < d(B, ∂B0).

The homotopies T and T provide a the desired “gain of regularity” as stated infollowing theorem.

Theorem 5.14. Let B = B(e, 1) and B′ = B(e, λ), λ > 1, be concentric balls of

Hn, and let 1 ≤ h ≤ 2n+ 1. If T, T are as in Theorem 5.12, then

i) T : W−1,p(B′, Eh+10 ) → Lp(B,Eh0 ) if h 6= n, and T : W−2,p(B,En+1

0 ) →Lp(B,En0 );

ii) T : Lp(B′, Eh0 )→ W 1,p(B,Eh−10 ), h 6= n+ 1, whereas T : Lp(B′, En+1

0 )→W 2,p(B,En0 ),

so that (40) still holds in Lp(B,E•0 ).In addition, for every (h, p, q) satisfying inequalities

1 < p ≤ q <∞, 1

p− 1

q≤

1Q if h 6= n+ 1,2Q if h = n+ 1,

(47)

we have:

iii) T : Lp(B′, Eh0 )→ Lq(B,Eh−10 ).

Proof. Let us prove i). Suppose h 6= n, and take β ∈ W−1,p(B′, Eh+10 ). As in

the proof of the previous theorem, let ψR be a cut-off function supported in a R-neighborhood of the origin, such that ψR ≡ 1 near the origin. Thus, again withthe notations of the proof of the previous theorem (see, in particular, (46) and

(42)), the operator K1,R is associated with a matrix-valued kernel ψR(k1)`,λ and β

is identified with a vector-valued distribution (β1, . . . , βNh), with βj =

∑iWif

ji as

in Remark 4.7, with∑j

∑i

‖f ji ‖Lp(B′) ≤ C‖β‖W−1,p(B′,Eh+10 ).

As in the the proof of previous theorem, let us fix two balls B0, B1 with B b B0 bB1 b B′. If χ ∈ D(B1) is cut-off function such that χ ≡ 1 on B0, we set β0 = χβ.Thus (β0)j , the j-th component of β0 has the form

(β0)j =∑i

Wi(χfji )−

∑i

(Wiχ)f ji .

Keeping in mind Remark 4.7, in order to estimate the norm of T β in Lp(B,Eh0 ),

we estimate 〈T β|φ〉, where

φ =∑j

φjξhj ∈ D(B,Eh0 ), with ‖φ‖Lp′ (B′,Eh

0 ) ≤ 1.


By (29), 〈T β|φ〉 is a sum of terms of the form

(48)

∫B

(ψRκ ∗ f0)(x)Wiφ(x) dx = 〈ψRκ ∗Wif0|φ〉

(where, as above, f0 = χf), or of the form

(49)

∫B

(ψRκ ∗ (Wiχ)f)(x)φ(x) dx,

where κ denotes one of the kernels (k1)`,λ of type 1 associated with k1 (see (41) in

the proof of previous theorem), f is one of the f ji ’s and φ one of the φj ’s,As for (48), by (23),

〈ψRκ ∗Wif0|φ〉 = 〈vW I v[ψRκ] ∗ f0|φ〉= 〈ψRvW I vκ ∗ f0|φ〉 − 〈(vW I vψR)κ ∗ f0|φ〉.

We notice now that vW I vκ is a kernel of type 0. Therefore, by Lemma 3.5 (keepin mind that f0 and φ are real function)

〈ψRvW I vκ ∗ f0|φ〉 ≤ ‖ψRvW I vκ ∗ f0‖Lp(B)‖φ‖Lp′ (B)

≤ ‖ψRvW I vκ ∗ f0‖Lp(B) ≤ C‖f0‖Lp(B′)

≤ C‖β‖W−1,p(B′,Eh+10 ).

The term in (49) can be handled in the same way, keeping into account Remark(3.6). Eventually, combining (48) and (49) we obtain that

‖T β‖Lp(B,Eh0 ) ≤ C‖β‖W−1,p(B′,Eh+1

0 ).

The assertion for h = n can be proved in the same way, taking into account thatT is built from a kernel of type 2, and that the norm in the space W−2,p(B,En+1

0 ) isexpressed by duality in terms of second order horizontal derivatives of test functions(see Remark 4.5).

Let us prove now ii). Suppose h 6= n + 1 and take α =∑j αjξ

hj ∈ D(B′, Eh0 ).

Arguing as above, in order to estimate ‖Tα‖W 1,p(B,Eh−10 ) we have to consider terms

of the form

(50) W`(ψRκ ∗ (χαj)) = ψRκ ∗ (W`(χαj))

(when we want to estimate the Lp-norm of the horizontal derivatives of Tα), or ofthe form

(51) ψRκ ∗ (χαj)

(when we want to estimate the Lp-norm of Tα). Both (50) and (51) can be handledas in the case i) (no need here of the duality argument).

We point out that (51) yields a Lp−Lq estimates (since, unlike (50), it involvesonly kernels of type 1) and then assertion iii) follows.

The operator S is the required local smoothing operator. More precisely, wehave:

Theorem 5.15. Let B = B(e, 1) and B′ = B(e, λ), λ > 1, be concentric balls ofHn, and let 1 ≤ h ≤ 2n + 1. Then the operator S defined in (46) is a smoothingoperator. In particular, for any m, s ∈ Z, m < s, S is bounded from Wm,p(B′, Eh0 )to W s,q(B,Eh0 ) for any p, q ∈ (1,∞) and maps Wm,p(B′, Eh0 ) into C∞(B,Eh0 ).


Proof. Since B is bounded, we can assume q > p. First take m = 0. Again, let usfix two balls B0, B1 with B b B0 b B1 b B′. If χ ∈ D(B1) is cut-off function suchthat χ ≡ 1 on B0, we set α0 = χα. Keeping the notations of the proof of Theorem5.12, it is easy to check that Sα can be written as (see (45))

(52) Sα = S0α0 := α0 ∗ dc(1− ψR)k1)± α0 ∗ vdcv(1− ψR)k1.

Thus, if α =∑j αjξ

hj , then each entry of Sα is a sum of terms of the form

(χαj) ∗ κ,

where κ ia a smooth kernel. Thus we are lead to estimate the Lq-norms in B of asum of terms of the form

(χαj) ∗W Jκ = (χαj) ∗ 12B′WJκ with |J | = s,

and the assertion follows by classical Hausdorff-Young inequality (see [20], Propo-sition 1.10 ), since the kernel 12B′W

Jκ belongs to all Lr, r ≥ 1. Therefore S isbounded from Lp(B′, Eh0 ) to W s,q(B,Eh0 ). Clearly, this yields the continuity of Sfrom Wm,p(B′, Eh0 ) to W s,q(B,Eh0 ) for m ≥ 0.

The proof in the case m < 0 can be carried out by a duality argument akin tothe one we used in the proof of Theorem 5.14.

Remark 5.16. Apparently, in previous theorem, two different homotopy operatorsT and T appear. In fact, they coincide when acting on form of the same degree.

More precisely, in Proposition 5.9 the homotopy formulas involve four operatorsK1, K1,K2, K2, where the notation is meant to distinguish operators acting on dcα(the operators with tilde) from those on which the differential acts (the operatorswithout tilde), whereas the lower index 1 or 2 denotes the type of the associatedkernels. Alternatively, a different notation could be used: if α ∈ D(Hn, Eh0 ) we canwrite

α = dcKh + Kh+1dcα,

where the tilde has the same previous meaning, whereas the lower index refers nowto the degree of the forms on which the operator acts.

It is important to notice that

Kh+1 = Kh+1, h = 1, . . . , 2n.

Indeed, take h < n − 1. Then Kh+1 = δc∆−1H,h+1 (as it appears in the homotopy

formula at the degree h), that equals Kh+1 (as it appears in the homotopy formula

at the degree h + 1 ≤ n − 1). Take now h = n − 1. Then Kn = δcdcδc∆−1H,n (as

it appears in the homotopy formula at the degree n), that equals Kn (as it appears

in the homotopy formula at the degree n). If h = n, then Kn+1 = δc∆−1H,n+1 (as it

appears in the homotopy formula at the degree n), that equals Kn+1 (as it appears in

the homotopy formula at the degree n+1). Finally, if h > n, then Kh+1 = δc∆−1H,h+1

(as it appears in the homotopy formula at the degree h), that equals Kh+1 (as itappears in the homotopy formula at the degree h+ 1).

Once this point is established, from now on we shall write

K := Kh = Kh

without ambiguity.


Therefore T = T and the homotopy formula (40) reads as

(53) dcT + Tdc + S = I on B.

It is worth pointing out the following fact.

Remark 5.17. As above, take B := B(e, 1), B′ := B(e, λ) with λ > 1, and, as informula (43), let us fix two balls B0, B1 with

B b B0 b B1 b B′,

and a cut-off function χ ∈ D(B1), χ ≡ 1 on B0.Take now α, β ∈ Lp(B′, E•0 ), α ≡ β on B1. Then, by (53), Sα = Sβ in B.

Indeed, if α0 := χα ≡ β0 := χβ in B0, then K1,Rα0 ≡ K1,Rβ0 and K1,Rdcα0 ≡K1,Rdcβ0 in B. In other words, (dcT + Tdc)α = (dcT + Tdc)β in B.

The following commutation lemma will be helpful in the sequel.

Lemma 5.18. As above, take B := B(e, 1), B′ := B(e, λ) with λ > 1, and, as informula (43), let us fix two balls B0, B1 with

(54) B b B0 b B1 b B′,

and a cut-off function χ ∈ D(B1), χ ≡ 1 on B0.We have:

Sdcα = dcSα for all α ∈ Lp(Hn, Eh0 ),

1 ≤ h ≤ 2n+ 1.

Proof. By (53), if α ∈ D(B′, Eh0 ), then Sdcα = dcSα. The case α ∈ Lp(B′, Eh0 )requires more technicalities.

Indeed, take α ∈ Lp(B′, Eh0 ), and let χ1 be a cut-off function supported in B′,χ1 ≡ 1 on B1. By convolution with usual Friedrichs’ mollifiers (see Definition 3.1),we can find a sequence (αk)k∈N in D(B′, Eh0 ) converging to χ1α in Lp(B′, Eh0 ). By

Theorem 5.14, Sαk → S(χ1α) in W 2,p(B,Eh+10 ), and hence dcSαk → dcS(χ1α) in

Lp(B,Eh0 ) as k →∞.On the other hand, χ1α ≡ α in B1, and then by Remark 5.17 S(χ1α) = Sα in

B, so that dcSαk → dcSα in Lp(B,Eh0 ) as k →∞.In addition, dcαk → dc(χ1α) in W−1,p(B′, Eh0 ) (in W−2,p(B′, Eh0 ) if h = n) and

hence, by Theorem 5.12, Sdcαk → Sdc(χ1α) in B as k →∞. Again dc(χ1α) ≡ dcαin B1 and then, by Remark 5.17, Sdcαk → Sdcα in B as k →∞.

Finally, since dcSαk = Sdcαk for all k ∈ N, we can take the limits as k → ∞and the assertion follows.

The following theorem contains one of the main results of the paper: it yieldsinterior Poincare inequality and Sobolev inequality for Rumin forms in the sense ofDefinitions 5.1 and 5.5.

Theorem 5.19. Take λ > 1 and set B = B(e, 1) and B′ = B(e, λ). If 1 ≤ h ≤2n+ 1, as in (47), take

1 < p ≤ q <∞, 1

p− 1

q≤

1Q if h 6= n+ 1,2Q if h = n+ 1.

(55)

Then


i) an interior H-Poincarep,q(h) inequality holds with respect to the balls B andB′.

ii) In addition, an interior H-Sobolevp,q(h) inequalities holds for 1 ≤ h ≤ 2n.

Proof. i) Interior H-Poincarep,q(h) inequality: let ω ∈ Lp(B′, Eh0 ) be dc-closed. By(53) we can write

(56) ω = dcTω + Sω in B.

By Theorem 5.15, we have Sω ∈ C∞(B,Eh0 ). Furthermore, dcSω = 0 since dcω =d2cTω + dcSω in B and dcω = 0 (by assumption).

Thus we can apply (38) to Sω and we get Sω = dcKSω, where K is defined in(36). In B, put now

φ := (KS + T )ω.

Trivially dcφ = dcKSω+ dcTω = Sω+ dcTω = ω, by (56). By Theorems 5.14 and5.15,

‖φ‖Lq(B,Eh−10 ) ≤ ‖KSω‖Lq(B,Eh−1

0 ) + ‖Tω‖Lq(B,Eh−10 )

≤ ‖KSω‖Lq(B,Eh−10 ) + C‖ω‖Lp(B′,Eh

0 )

≤ C(‖Sω‖W 1,q(B,Eh

0 ) + ‖ω‖Lp(B′,Eh0 )

)(by Lemma 5.8)

≤ C‖ω‖Lp(B′,Eh0 ).

(57)

ii) Interior H-Sobolevp,q(h) inequality: as in formula (43), let us fix two balls B0,B1 with

B b B0 b B1 b B′,

and a cut-off function χ ∈ D(B1), χ ≡ 1 on B0.Let ω ∈ Lp(B,Eh0 ) be a compactly supported form such that dcω = 0. Since ω

vanishes in a neighborhood of ∂B, without loss of generality we can assume that itis continued by zero on B′. In addition, ω = χω, since χ ≡ 1 on suppω.

By (53) we have ω = dcTω + Sω. On the other hand, since ω vanishes outsideB, by its very definition (see (46)) Tω is supported in B0 by Remark 5.13, so thatalso Sω is supported in B0.

Again as above Sω ∈ C∞(B,Eh0 ), and dcSω = 0. Thus we can apply (38) to Sωand we get Sω = dcJSω, where J is defined in (37) (that preserves the support).By Lemma 5.7, JSω is supported in B0 ⊂ B′. Thus, if we set φ := (JS + T )ω,then φ is supported in B′. Moreover dcφ = dcKSω + dcTω = Sω + ω − Sω = ω.

At this point, we can repeat the estimates (57) and we get eventually

‖φ‖Lq(B′,Eh−10 ) ≤ C‖ω‖Lp(B,Eh

0 ).

This completes the proof of the theorem.

If p ∈ Hn and t > 0, then the map x → f(x) := τpδt(x) maps B(e, ρ) intoB(p, tρ) for ρ > 0. Therefore, by Proposition 2.12, from the previous theorem forballs of fixed radius, we obtain the following result for general balls.

Theorem 5.20. Take 1 ≤ h ≤ 2n + 1. Suppose 1 < p < Q if h 6= n + 1 and1 < p < Q/2 if h = n+ 1. Let q ≥ p such that

1

p− 1

q≤

1Q if h 6= n+ 1,2Q if h = n+ 1.

(58)


Then there exists a constant C such that, for every dc-closed differential h-formω in Lp(B(p, λr);Eh0 ) there exists a (h− 1)-form φ in Lq(B(p, r), Eh−1

0 ) such thatdcφ = ω and

(59) ‖φ‖Lq(B(p,r),Eh−10 ) ≤ C r

Q/q−Q/p+1 ‖ω‖Lp(B(p,λr),Eh0 ) if h 6= n+ 1

and

(60) ‖φ‖Lq(B(p,r),En0 ) ≤ C rQ/q−Q/p+2 ‖ω‖Lp(B(p,λr),En+1

0 ).

Analogously, if 1 ≤ h ≤ 2n there exists a constant C such that, for every compactlysupported dc-closed h-form ω in Lp(B(p, r);Eh0 ) there exists a compactly supported

(h− 1)-form φ in Lq(B(p, λr), Eh−10 ) such that dcφ = ω in B(p, λr) and

(61) ‖φ‖Lq(B(p,λr),Eh−10 ) ≤ C ‖ω‖Lp(B(p,r),Eh

0 )

Proof. We have just to take the pull-back f#ω and then apply Theorem 5.19.

If the choice of q is sharp (i.e. in (58) the equality holds), then the constant onthe right hand side of (61) is independent of the radius of the ball, so that a globalH-Sobolevp,q(h) inequality holds.

Therefore we get the following result.

Corollary 5.21. Take 1 ≤ h ≤ 2n. Suppose 1 < p < Q if h 6= n + 1 and1 < p < Q/2 if h = n+ 1. Let q ≥ p defined by

1

p− 1

q:=

1Q if h 6= n+ 1,2Q if h = n+ 1.

(62)

Then H-Sobolevp,q(h) inequality holds for 1 ≤ h ≤ 2n.

In the case H1, for 1-forms and 2-forms for instance, the primitive φ of a com-pactly supported form can be written explicitly as in Example 5.11.

Remark 5.22. A scaling argument shows easily that the exponents in (59) and (60)are sharp. On the other hand, we have already discussed in Section 1.5 whether sim-ilar sharp results can be proved for general Carnot groups, stating ultimately thatthis is not possible (at least relying on our present arguments). Now the argumentof Section 1.5 can be made more precise. If we look at the proofs of our inequali-ties, we see that at the very beginning there is an approximate homotopy formulathat in turn descends from the existence of a fundament solution for a suitablehypoelliptic homogeneous ”artificial Laplacian”. This construction is still possiblein general Carnot groups (see [8], [48]) relying on the construction of a ”0-orderLaplacian”, but the approximate homotopy formula involves singular integral oper-ators that fail to have the good homogeneity. This is due to the fact that in generalCarnot groups, with exception of very particular cases, the forms of a given degreein Rumin’s complex have different weights (this doesn’t happen in Euclidean spacesand in Heisenberg groups). We stress that this phenomenon appears already in step2 groups, very akin to Heisenberg groups, like quaternionic Heisenberg groups (see[11]), that are defined by replacing the complex field C by the field of quaternionsin the definition of H1. This generates a two-step Carnot group whose centre is3-dimensional (while the centre in Hn is 1-dimensional).

Thus the quaternionic Heisenberg group (in dimension 7) is a nilpotent Lie groupwith underlying manifold R4

x × R3t , where x = (x1, x2, x3, x4) and t = (t1, t2, t3).


A basis for the Lie algebra of left-invariant vector fields on the group is given by:

X1 =∂

∂x1+

1

2x2

∂

∂t1+

1

2x3

∂

∂t2+

1

2x4

∂

∂t3;

X2 =∂

∂x2−

1

2x1

∂

∂t1+

1

2x4

∂

∂t2−

1

2x3

∂

∂t3;

X3 =∂

∂x3−

1

2x4

∂

∂t1−

1

2x1

∂

∂t2+

1

2x2

∂

∂t3;

X4 =∂

∂x4+

1

2x3

∂

∂t1−

1

2x2

∂

∂t2−

1

2x1

∂

∂t3;

Tk =∂

∂tkfor k = 1, 2, 3 .

The non-trivial commutation relations are:

[X1, X2] = −[X3, X4] = −T1 ; [X1, X3] = [X2, X4] = −T2;[X1, X4] = −[X2, X3] = −T3 .

The standard quaternionic contact forms τ1, τ2, τ3 are given by:

τ1 = dt1 − 12 x2dx1 + 1

2 x1dx2 − 12 x4dx3 + 1

2 x3dx4;

τ2 = dt2 − 12 x3dx1 + 1

2 x4dx2 + 12 x1dx3 − 1

2 x2dx4;

τ1 = dt3 − 12 x4dx1 − 1

2 x3dx2 + 12 x2dx3 + 1

2 x1dx4 .

The space of intrinsic 1-forms and 2-forms are

E10 = Ω1,1 = spandx1, dx2, dx3, dx4,

and

E20 = spanα2, α4, α6 ⊕ spanβ1, β2, β3, β4, β5, β6, β7, β8 ,

where

α1 := dx1 ∧ dx2 + dx3 ∧ dx4 , α2 := dx1 ∧ dx2 − dx3 ∧ dx4 ,

α3 := dx1 ∧ dx3 − dx2 ∧ dx4 , α4 := dx1 ∧ dx3 + dx2 ∧ dx4 ,

α5 := dx1 ∧ dx4 + dx2 ∧ dx3 , α6 := dx1 ∧ dx4 − dx2 ∧ dx3 .

and

β1 := dx1 ∧ τ2 + dx4 ∧ τ1 , β2 := dx2 ∧ τ3 + dx4 ∧ τ1 , β3 := dx1 ∧ τ3 + dx2 ∧ τ2 ,β4 := dx3 ∧ τ1 + dx2 ∧ τ2 , β5 := dx1 ∧ τ1 + dx3 ∧ τ3 , β6 := dx4 ∧ τ2 + dx3 ∧ τ3 ,β7 := dx2 ∧ τ1 − dx4 ∧ τ3 , β8 := −dx3 ∧ τ2 + dx4 ∧ τ3 , β9 := dx1 ∧ τ2 − dx4 ∧ τ1 ,β10 := dx1 ∧ τ3 − dx2 ∧ τ2 , β11 := dx1 ∧ τ1 − dx3 ∧ τ3 , β12 := dx2 ∧ τ1 + dx4 ∧ τ3 ,respectively. It turns out that α2, α4, α6 have weight 2, whereas β1, . . . β10 haveweight 3.

6. Contact manifolds and global smoothing

Throughout this section, (M,H, g) will be a sub-Riemannian contact manifoldof bounded Ck-geometry as in Definition 4.9, k ≥ 3. We shall denote by (E•0 , dc)both the Rumin complex in (M,H, g) and in the Heisenberg group.

The core of this section consists in the proof of an approximate homotopy formula

(63) I = dcTM + TMdc + SM ,

where the “error term” SM has the maximal regularising property compatible withthe regularity of M , and TM enjoys the natural continuity properties between


Sobolev spaces on M . The proof will be carried out in two steps: first (Lemma 6.1)we shall prove an approximate homotopy formula akin to (63) where SM “gainsonly one horizontal derivative”, and then, iterating (63), we obtain the desiredapproximate homotopy formula, where SM has the maximal regularising propertycompatible with the regularity of M .

As in Definition 4.12, let now χj be a partition of the unity subordinate tothe atlas U := B(xj , ρ), φxj

of Lemma 4.11. From now on, for sake of simplicity,

we shall write φj := φxj. We stress again that φ−1

j (supp χj) ⊂ B(e, 1).

If u ∈ Lp(M,E•0 ), we have

u =∑j

χju.

We can write

χju = (φ−1j )#φ#

j (χju) =: (φ−1j )#vj .

We use now the homotopy formula in Hn (see Theorem 5.12):

vj = dcTvj + Tdcvj + Svj in B(e, 1).

Without loss of generality, we can assume that R > 0 in the definition of the kernelof T has been chosen in such a way that the R-neighborhood of φ−1

j (supp χj) ⊂B(e, 1). In particular vj − dcTvj − Tdcvj is supported in B(e, 1) and therefore alsoSvj is supported in B(e, 1).

In particular, (φ−1j )#

(dcTvj + Tdcvj + Svj

)is supported in φj(B(e, 1)) so that

it can be continued by zero on M .Thus

u =∑j

(φ−1j )#

(dcTvj + Tdcvj + Svj

)= dc

∑j

(φ−1j )#Tφ#

j (χju)

+∑j

((φ−1j )#Tφ#

j χj)dcu−∑j

(φ−1j )#Tφ#

j ([χj , dc]u)

+∑j

((φ−1j )#(Sφ#

j χj)u.

We set

(64) Tu :=∑j

(φ−1j )#Tφ#

j (χju)

and

(65) Su :=∑j

(φ−1j )#Sφ#

j (χju)−∑j

(φ−1j )#Tφ#

j ([χj , dc]u).

Lemma 6.1. Let (M,H, g) be a bounded Ck-geometry sub-Riemannian contactmanifold with k ≥ 3. If 2 ≤ ` ≤ k − 1 and T and S are defined in (64) and (65),then the following homotopy formula holds:

(66) I = dcT + Tdc + S.

In particular, Sdc = dcS. In addition, if 1 ≤ h ≤ 2n + 1, the following maps arecontinuous:


i) T : W−1,p(M,Eh+10 )→ Lp(M,Eh0 ) if h 6= n, whereas T : W−2,p(M,En+1

0 )→Lp(M,En0 );

ii) T : Lp(M,Eh0 )→W 1,p(M,Eh−10 ) if h 6= n+1, whereas T : Lp(M,En+1

0 )→W 2,p(M,En0 );

iii) if 1 ≤ ` ≤ k, then S : W `−1,p(M,Eh0 ) −→W `,p(M,Eh0 ).

Proof. First of all, we notice that, if α is supported in φj(B(e, λ)), then, by Defi-nition 4.9 the norms

‖α‖Wm,p(M,E•0 ) and ‖φ#j α‖Wm,p(Hn,E•0 )

are equivalent for −k ≤ m ≤ k, with equivalence constants independent of j. Thus,assertions i) and ii) follow straightforwardly from Theorem 5.14.

To get iii) we only need to note that the operators (φ−1j )#Tφ#

j [χj , dc] are

bounded W `−1,p(M,E•0 ) → W `,p(M,E•0 ) in every degree. Indeed, by Proposi-

tion 2.14, the differential operator φ#j [χj , dc](φ

−1j )# in Hn has order 1 if h = n,

and order 0 if h 6= n. Since the kernel of T can be estimated by kernel of type 2 ifT acts on forms of degree h = n, and of type 1 if it acts on forms of degree h 6= n,the assertion follows straightforwardly.

Summing up in j and keeping into account that the sum is locally finite, weobtain:

‖∑j

(φ−1j )#Tφ#

j [χj , dc]u‖W `,p(M,E•0 ) ≤∑j

‖(φ−1j )#Tφ#

j [χj , dc]u‖W `,p(φj(B(e,1)),E•0 )

≤ C∑j

‖Tφ#j [χj , dc]u‖W `,p(B(e,1),E•0 ) ≤ C

∑j

‖φ#j u‖W `−1,p(B(e,1),E•0 )

≤ C‖u‖W `−1,p(M,E•0 ).

Now the following global homotopy formula holds in M .

Theorem 6.2. Let (M,H, g) be a bounded Ck-geometry sub-Riemannian contactmanifold, k ≥ 3. Then

(67) I = dcTM + TMdc + SM ,

where

TM :=( k−1∑i=0

Si)T, SM := Sk,

and T and S are defined in (64) and (65).Moreover

(68) dcSMu = SMdcu,

and, if 1 ≤ h ≤ 2n+ 1, the following maps are continuous:

i) TM : W−1,p(M,Eh+10 )→ Lp(M,Eh0 ) if h 6= n, whereas TM : W−2,p(M,En+1

0 )→Lp(M,En0 );

ii) TM : Lp(M,Eh0 )→W 1,p(M,Eh−10 ) if h 6= n+1, whereas TM : Lp(M,En+1

0 )→W 2,p(M,En0 );

iii) SM : Lp(M,Eh0 )→W k−1,p(M,Eh0 ).


Proof. By (68),

dcTM + TMdc + SM

= dc( k−1∑i=0

Si)T +

( k−1∑i=0

Si)T dc + Sk

=

k−1∑i=0

Si(dcT + Tdc

)+ Sk

=

k−1∑i=0

Si(I − S) + Sk = I.

Then statements i), ii) and iii) follow straightforwardly from i), ii) and iii) of Lemma6.1.

7. Large scale geometry of contact sub-Riemannian manifolds

Theorems 1.2 and 1.5 are the key to proving that the validity of global Poincareinequalities is equivalent to vanishing of `q,p cohomology, a large scale invariant ofmetric spaces. This equivalence will be established in [47]. By large scale invariant,we mean preserved, under uniform local assumptions, by quasiisometries, i.e. mapsf between metric spaces which satisfy

−C +1

Ld(x, x′) ≤ d(f(x), f(x′)) ≤ Ld(x, x′) + C,

for suitable positive constants L and C.Avoiding the general metric definition of `q,p cohomology, let us give a construc-

tion valid for bounded geometry Riemannian manifolds with uniform vanishing ofcohomology (the cohomology of an R′-ball dies when restricted to a concentric R-ball, where the radius R′ depends only on the radius R). First, one defines the `q,p

cohomology of a simplicial complex: it is the quotient of the space of `p simplicialcocycles by the image of `q simplicial cochains by the coboundary operator. Oneshows that `q,p cohomology is a quasiisometry invariant of simplicial complexeswith bounded geometry (i.e. bounded number of simplices through a vertex) anduniform vanishing of cohomology. Then one observes that every bounded geometryRiemannian manifold is quasiisometric to such a simplicial complex.

Under similar boundedness and uniformity assumptions, one can show ([47])that various locally acyclic complexes can be used to compute `q,p cohomology. Forcontact sub-Riemannian manifolds, one can use either the exterior differential orRumin’s differential. As alluded to above, the building blocks are interior estimatesand global smoothing, i.e. Theorems 1.2 and 1.5 and their Riemannian analogues.It follows that a global Poincare inequality holds if and only if a global H-Poincareinequality holds.

Using the Riemannian Hodge Laplacian, D. Muller, M. Peloso and F. Ricci provea Poincare inequality Poincare2,q for the exterior differential on the RiemannianHeisenberg group ([44], Lemma 11.2), under the assumption 1

2 −1q = 1

2n+1 . There-

fore, their result combined with [47] provides an alternative proof of part of Corol-lary 1.4 above. We note that in degree h = n + 1, they miss the sharp exponent,given by our condition E(n+ 1, 2, q, n).


The advantage of Rumin’s Laplacian over its Riemannian sibling is its scaleinvariance. This allows to apply the theory of singular integral operators, to treat`q,p cohomology for all p and to get the sharp exponent in degree h = n + 1. Thedrawback of Rumin’s complex is that interior Poincare inequalities become hard.

7.1. Three-dimensional Lie groups. There are four 3-dimensional Lie algebraswhich cannot be generated by a pair of vectors: the abelian Lie algebra R3, dil(2),the direct sum dil(1) ⊕ R, where dil(n) denotes the Lie algebra of the group ofdilations and translations of Rn, and the solvable unimodular Lie algebra sol. TheLie groups corresponding to other 3-dimensional Lie algebras admit left-invariantcontact structures. All left-invariant sub-Riemannian metrics have bounded geom-etry, so Theorem 1.5 applies. When simply connected, they satisfy all uniform localassumptions required for identification of H-Poincarep,q inequality with vanishingof `q,p cohomology and its quasiisometry invariance. Here are examples.

Heisenberg group H1 is covered by Theorem 1.1. Note that the correspondingfacts about `q,p cohomology are new.

M1 := Mot(E2), the universal covering of the group of planar Euclidean motions,is quasiisometric to Euclidean 3-space E3. Its `q,p cohomology vanishes if and onlyif 1p −

1q ≥

13 (this is the Euclidean analogue of Theorem 1.1). Therefore, assuming

1 < p ≤ q < ∞, the H-Poincarep,q inequality holds for this group if and only if1p −

1q ≥

13 , in all degrees.

M2 := ˜SL(2,R), the universal covering of SL(2,R), is quasiisometric to PSL(2,R)×R. In degree 1, its `p,p-cohomology vanishes for all p > 1, see [46]. Since PSL(2,R)acts isometrically and simply transitively on hyperbolic plane H2, it is quasiisomet-ric to H2. Since the `p,p-cohomology of H2 in degree 1 is Hausdorff and nonzero,the Kunneth formula of [29] applies, and the `p,p-cohomology in degree 2 of theproduct does not vanish, because the `p,p-cohomology in degree 1 of the line doesnot vanish. We conclude that, assuming 1 < p < ∞, the H-Poincarep,p inequalityholds in degree 1, and only in degree 1.

7.2. Other examples. Next we describe a few non simply connected examples.Then the quasiisometry invariance holds only in degree 1.

Let M0 be the quotient of Heisenberg group H1 by the discrete subgroup Γgenerated by two elements, one of which belongs to the center of H1. Let us equipit with the quotient contact structure and sub-Riemannian metric. Γ is containedin a connected subgroup L of H1 isomorphic to R2. This gives rise to a fibrationM0 → L \ H1, which is a line. The fibers of this map are tori with uniformlybounded diameters, therefore it is a quasiisometry. The `q,p cohomology of theline is well understood, it vanishes only when (q, p) = (∞, 1). Therefore, assuming1 < p ≤ q <∞, the H-Poincarep,q inequality never holds for M0 in degree 1.

Let M1 denote the unit cotangent bundle of Euclidean plane E2. It carriesa tautological contact structure. The group G1 of motions of Euclidean plane,which is a semi-direct product of R2 with SO(2), acts simply transitively on M ,preserving the contact structure. Pick a G1-invariant sub-Riemannian metric onM . By invariance, the bounded geometry assumption is satisfied. The projectionM → E2 has uniformly bounded fibers, it is a quasiisometry. Therefore M and E2

have isomorphic exact `q,p cohomologies in degree 1. The `q,p cohomology of E2 iswell understood. It vanishes if and only if 1

p −1q ≥

12 . We conclude that, assuming


1 < p ≤ q < ∞, the H-Poincarep,q inequality holds for M1 in degree 1 if and onlyif 1p −

1q ≥

12 .

Let us replace Euclidean plane with hyperbolic plane H2. The construction isidentical, up to the structure of the identity component G2 of the isometry groupof hyperbolic plane: it is isomorphic to PSL(2,R). The obtained sub-Riemannianmanifold M2 is quasiisometric to H2. The `q,p cohomology of H2 in degree 1 iswell understood. It vanishes only for p = 1. We conclude that the H-Poincarep,qinequality never holds in degree 1 for M2 if 1 < p ≤ q <∞.

7.3. Further remarks. In each degree k, for every p, there is an exponent q =q(n, k) such that the Lq-norm of Rumin k-forms is a conformal invariant (q(n, k) =2n+2k if k ≤ n, q(n, k) = 2n+2

k+1 if k ≥ n+ 1). Therefore, in degree k, `q(n,k−1),q(n,k)

cohomology of 2n+1-dimensional contact sub-Riemannian manifolds is a quasicon-formal invariant, and so does the validity of a H-Poincareq(n,k),q(n,k−1) inequality.We note that if k < 2n + 1, for Heisenberg group Hn, these cohomology groupsvanish, whereas they need not vanish for other examples. For instance, if n = 1,

q(n, 1) = 4, q(n, 2) = 2, `4,2-cohomology in degree 2 of M1 does not vanish. This

shows that M1 is not quasiconformally equivalent to H1.We see that Theorems 1.2 and 1.5 constitute useful tools for the geometric study

of mappings between contact sub-Riemannian manifolds. Here are a few referencesabout this emerging subject: [35] shows that two ways to take a quotient of Heisen-berg group by an isometry give rise to contact sub-Riemannian manifolds whichare not quasiconformal. Moreover [31] establish the basic properties of quasiregu-lar maps, a study which has been continued in [37], [12], [19].

Acknowledgments

The authors are happy to thank the referees for their comments and suggestionsthat improved the readability of the paper.

A.B. and B.F. are supported by the University of Bologna, funds for selectedresearch topics, and by MAnET Marie Curie Initial Training Network, by GNAMPAof INdAM (Istituto Nazionale di Alta Matematica “F. Severi”), Italy, and by PRINof the MIUR, Italy.

P.P. is supported by MAnET Marie Curie Initial Training Network, by AgenceNationale de la Recherche, ANR-10-BLAN 116-01 GGAA and ANR-15-CE40-0018SRGI. P.P. gratefully acknowledges the hospitality of Isaac Newton Institute, ofEPSRC under grant EP/K032208/1, and of Simons Foundation.

References

1. Annalisa Baldi, Marilena Barnabei, and Bruno Franchi, A recursive basis for primitive forms

in symplectic spaces and applications to Heisenberg groups, Acta Math. Sin. (Engl. Ser.) 32(2016), no. 3, 265–285. MR 3456421

2. Annalisa Baldi and Bruno Franchi, Sharp a priori estimates for div-curl systems in Heisenberg

groups, J. Funct. Anal. 265 (2013), no. 10, 2388–2419. MR 30918193. Annalisa Baldi, Bruno Franchi, and Pierre Pansu, Gagliardo-Nirenberg inequalities for differ-

ential forms in Heisenberg groups, Math. Ann. 365 (2016), no. 3-4, 1633–1667. MR 3521101

4. , L1-Poincare and Sobolev inequalities for differential forms in Euclidean spaces, Sci.

China Math. 62 (2019), no. 6, 1029–1040. MR 3951879

5. , in preparation, 2020.6. , L1-Poincare inequalities for differential forms on Euclidean spaces and Heisenberg

groups, Adv. Math. 366 (2020), 107084. MR 4070308


7. , Orlicz spaces and endpoint Sobolev-Poincare inequalities for differential forms in

Heisenberg groups, Matematiche (Catania) 75 (2020), 167–194.

8. Annalisa Baldi, Bruno Franchi, Nicoletta Tchou, and Maria Carla Tesi, Compensated com-pactness for differential forms in Carnot groups and applications, Adv. Math. 223 (2010),

no. 5, 1555–1607.

9. Annalisa Baldi, Bruno Franchi, and Maria Carla Tesi, Compensated compactness in the con-tact complex of Heisenberg groups, Indiana Univ. Math. J. 57 (2008), 133–186.

10. , Hypoellipticity, fundamental solution and Liouville type theorem for matrix–valued

differential operators in Carnot groups, J. Eur. Math. Soc. 11 (2009), no. 4, 777–798.11. Annalisa Baldi, Bruno Franchi, and Francesca Tripaldi, Gagliardo-Nirenberg inequalities for

horizontal vector fields in the Engel group and in the 7-dimensional quaternionic Heisenberg

group, Geometric Methods in PDE’s, Springer INdAM Ser., vol. 13, Springer, 2015, pp. 287–312.

12. Zoltan M. Balogh, Katrin Fassler, and Kirsi Peltonen, Uniformly quasiregular maps on thecompactified Heisenberg group, J. Geom. Anal. 22 (2012), no. 3, 633–665. MR 2927672

13. Andreas Bernig, Natural operations on differential forms on contact manifolds, Differential

Geom. Appl. 50 (2017), 34–51. MR 358863914. Andrea Bonfiglioli, Ermanno Lanconelli, and Francesco Uguzzoni, Stratified Lie groups and

potential theory for their sub-Laplacians, Springer Monographs in Mathematics, Springer,

Berlin, 2007. MR MR236334315. Jean Bourgain and Haım Brezis, New estimates for elliptic equations and Hodge type systems,

J. Eur. Math. Soc. (JEMS) 9 (2007), no. 2, 277–315. MR 2293957 (2009h:35062)

16. Robert Bryant, Michael Eastwood, A. Rod Gover, and Katharina Neusser, Some differentialcomplexes within and beyond parabolic geometry, arXiv:1112.2142, 2011.

17. Luca Capogna, Donatella Danielli, and Nicola Garofalo, Subelliptic mollifiers and a basic

pointwise estimate of Poincare type, Math. Z. 226 (1997), no. 1, 147–154. MR 147214518. Sagun Chanillo and Jean Van Schaftingen, Subelliptic Bourgain-Brezis estimates on groups,

Math. Res. Lett. 16 (2009), no. 3, 487–501. MR 2511628 (2010f:35042)19. Katrin Fassler, Anton Lukyanenko, and Kirsi Peltonen, Quasiregular mappings on sub-

Riemannian manifolds, J. Geom. Anal. 26 (2016), no. 3, 1754–1794. MR 3511457

20. Gerald B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark.Mat. 13 (1975), no. 2, 161–207. MR MR0494315 (58 #13215)

21. Gerald B. Folland and Elias M. Stein, Hardy spaces on homogeneous groups, Mathematical

Notes, vol. 28, Princeton University Press, Princeton, N.J., 1982. MR MR657581 (84h:43027)22. Bruno Franchi, Cristian E. Gutierrez, and Richard L. Wheeden, Weighted Sobolev-Poincare

inequalities for Grushin type operators, Comm. Partial Differential Equations 19 (1994), no. 3-

4, 523–604. MR 126580823. Bruno Franchi, Guozhen Lu, and Richard L. Wheeden, Representation formulas and weighted

Poincare inequalities for Hormander vector fields, Ann. Inst. Fourier (Grenoble) 45 (1995),

no. 2, 577–604. MR 1343563 (96i:46037)24. , A relationship between Poincare-type inequalities and representation formulas in

spaces of homogeneous type, Internat. Math. Res. Notices (1996), no. 1, 1–14. MR 1383947(97k:26012)

25. Bruno Franchi, Raul Serapioni, and Francesco Serra Cassano, Meyers-Serrin type theorems

and relaxation of variational integrals depending on vector fields, Houston J. Math. 22 (1996),no. 4, 859–890. MR 1437714

26. , Regular submanifolds, graphs and area formula in Heisenberg groups, Adv. Math.

211 (2007), no. 1, 152–203. MR MR2313532 (2008h:49030)27. Bruno Franchi and Raul Paolo Serapioni, Intrinsic Lipschitz graphs within Carnot groups, J.

Geom. Anal. 26 (2016), no. 3, 1946–1994. MR 351146528. Sylvestre Gallot, Dominique Hulin, and Jacques Lafontaine, Riemannian geometry, third ed.,

Universitext, Springer-Verlag, Berlin, 2004. MR 2088027

29. V. M. Gol’dshtein, V. I. Kuz’minov, and I. A. Shvedov, Lp-cohomology of warped cylinders,Sibirsk. Mat. Zh. 31 (1990), no. 6, 55–63. MR 1097955

30. Mikhael Gromov, Carnot-Caratheodory spaces seen from within, Sub-Riemannian geometry,

Progr. Math., vol. 144, Birkhauser, Basel, 1996, pp. 79–323. MR MR1421823 (2000f:53034)31. Juha Heinonen and Ilkka Holopainen, Quasiregular maps on Carnot groups, J. Geom. Anal.

7 (1997), no. 1, 109–148. MR 1630785


32. Bernard Helffer and Jean Nourrigat, Hypoellipticite maximale pour des operateurs polynomes

de champs de vecteurs, Progress in Mathematics, vol. 58, Birkhauser Boston Inc., Boston,

MA, 1985. MR MR897103 (88i:35029)33. Tadeusz Iwaniec and Adam Lutoborski, Integral estimates for null Lagrangians, Arch. Ratio-

nal Mech. Anal. 125 (1993), no. 1, 25–79. MR MR1241286 (95c:58054)

34. David Jerison, The Poincare inequality for vector fields satisfying Hormander’s condition,Duke Math. J. 53 (1986), no. 2, 503–523. MR MR850547 (87i:35027)

35. Youngju Kim, Quasiconformal conjugacy classes of parabolic isometries of complex hyperbolic

space, Pacific J. Math. 270 (2014), no. 1, 129–149. MR 324585136. Loredana Lanzani and Elias M. Stein, A note on div curl inequalities, Math. Res. Lett. 12

(2005), no. 1, 57–61. MR 2122730 (2005m:58001)

37. Anton Lukyanenko, Geometric mapping theory of the Heisenberg group, sub-Riemannianmanifolds, and hyperbolic spaces, ProQuest LLC, Ann Arbor, MI, 2014, Thesis (Ph.D.)–

University of Illinois at Urbana-Champaign. MR 332203538. Pierre Maheux and Laurent Saloff-Coste, Analyse sur les boules d’un operateur sous-elliptique,

Math. Ann. 303 (1995), no. 4, 713–740. MR 1359957 (96m:35049)

39. Jean Martinet, Formes de contact sur les varietes de dimension 3, pp. 142–163. Lecture Notesin Math., Vol. 209, Springer, Berlin, 1971. MR 0350771

40. Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in

Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995, Fractals andrectifiability. MR 1333890

41. Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology, second ed., Oxford

Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998.MR 1698616

42. Dorina Mitrea, Marius Mitrea, and Sylvie Monniaux, The Poisson problem for the exterior

derivative operator with Dirichlet boundary condition in nonsmooth domains, Commun. PureAppl. Anal. 7 (2008), no. 6, 1295–1333. MR 2425010

43. Richard Montgomery, A tour of subriemannian geometries, their geodesics and applications,Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, Providence,

RI, 2002. MR MR1867362 (2002m:53045)

44. Detlef Muller, Marco M. Peloso, and Fulvio Ricci, Analysis of the Hodge Laplacian on theHeisenberg group, Mem. Amer. Math. Soc. 233 (2015), no. 1095, vi+91. MR 3289035

45. Pierre Pansu, Metriques de Carnot-Caratheodory et quasiisometries des espaces symetriques

de rang un, Ann. of Math. (2) 129 (1989), no. 1, 1–60. MR 979599 (90e:53058)46. , Cohomologie Lp en degre 1 des espaces homogenes, Potential Anal. 27 (2007), no. 2,

151–165. MR 232250347. , Cup-products in lq,p-cohomology: discretization and quasi-isometry invariance, arXiv

1702.04984, 2017.

48. Pierre Pansu and Michel Rumin, On the `q,p cohomology of Carnot groups, Ann. H. Lebesgue1 (2018), 267–295. MR 3963292

49. Pierre Pansu and Francesca Tripaldi, Averages and the `q,1 cohomology of heisenberg groups,

Annales Mathematiques Blaise Pascal 26 (2019), no. 1, 81–100 (en).50. Michel Rumin, Formes differentielles sur les varietes de contact, J. Differential Geom. 39

(1994), no. 2, 281–330. MR MR1267892 (95g:58221)51. , Differential geometry on C-C spaces and application to the Novikov-Shubin numbers

of nilpotent Lie groups, C. R. Acad. Sci. Paris Ser. I Math. 329 (1999), no. 11, 985–990.

MR MR1733906 (2001g:53063)

52. , Sub-Riemannian limit of the differential form spectrum of contact manifolds, Geom.Funct. Anal. 10 (2000), no. 2, 407–452. MR MR1771424 (2002f:53044)

53. , Around heat decay on forms and relations of nilpotent Lie groups, Seminaire deTheorie Spectrale et Geometrie, Vol. 19, Annee 2000–2001, Semin. Theor. Spectr. Geom.,vol. 19, Univ. Grenoble I, 2001, pp. 123–164. MR MR1909080 (2003f:58062)

54. , An introduction to spectral and differential geometry in Carnot-Caratheodory spaces,Rend. Circ. Mat. Palermo (2) Suppl. 75 (2005), 139–196. MR MR2152359 (2006g:58053)

55. Gunter Schwarz, Hodge decomposition—a method for solving boundary value problems, Lec-

ture Notes in Mathematics, vol. 1607, Springer-Verlag, Berlin, 1995. MR MR1367287(96k:58222)


56. Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory in-

tegrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ,

1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III.MR MR1232192 (95c:42002)

57. Nicholas Th. Varopoulos, Laurent Saloff-Coste, and Thierry Coulhon, Analysis and geom-

etry on groups, Cambridge Tracts in Mathematics, vol. 100, Cambridge University Press,Cambridge, 1992. MR MR1218884 (95f:43008)

58. Frank W. Warner, Foundations of differentiable manifolds and Lie groups, Graduate Texts in

Mathematics, vol. 94, Springer-Verlag, New York-Berlin, 1983, Corrected reprint of the 1971edition. MR 722297

Annalisa Baldi and Bruno FranchiUniversita di Bologna, Dipartimento di MatematicaPiazza di Porta S. Donato 5, 40126 Bologna, Italy.e-mail: [email protected], [email protected].

Pierre PansuUniversite Paris-Saclay, CNRS, Laboratoire de mathematiques d’Orsay91405, Orsay, France.e-mail: [email protected]

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